Symbolism of genetics
Symbolism is a list and explanation of conventional names and terms used in any branch of science.
The foundations of genetic symbolism were laid by Gregor Mendel, who used alphabetic symbolism to designate traits. Dominant traits were designated by capital letters of the Latin alphabet A, B, C, etc., recessive- in small letters - a, b, c, etc. Letter symbolism, proposed by Mendel, is essentially an algebraic form of expressing the laws of inheritance of characteristics.
The following symbolism is used to indicate crossing.
Parents are designated by the Latin letter P (Parents - parents), then their genotypes are written down next to them. Female denoted by the symbol ♂ (mirror of Venus), male- ♀ (shield and spear of Mars). An “x” is placed between the parents to indicate crossing. The female genotype is written in first place, and the male in second.
First byknee designated F1 (Filli - children), the second generation - F2, etc. The designations of the genotypes of the descendants are given nearby.
Glossary of basic terms and concepts
Alternative signs– mutually exclusive, contrasting features.
Gametes(from Greek " gametes"- spouse) is a reproductive cell of a plant or animal organism that carries one gene from an allelic pair. Gametes always carry genes in a “pure” form, since they are formed by meiotic cell division and contain one of a pair of homologous chromosomes.
Gene(from Greek " genos"- birth) is a section of a DNA molecule that carries information about the primary structure of one specific protein.
Allelic genes– paired genes located in identical regions of homologous chromosomes.
Genotype- a set of hereditary inclinations (genes) of an organism.
Heterozygote(from Greek " heteros" - other and zygote) - a zygote that has two different alleles for a given gene ( Aa, Bb).
Homozygote(from Greek " homos" - identical and zygote) - a zygote that has the same alleles of a given gene (both dominant or both recessive).
Homologous chromosomes(from Greek " homos" - identical) - paired chromosomes, identical in shape, size, set of genes. In a diploid cell, the set of chromosomes is always paired: one chromosome is from a pair of maternal origin, the second is of paternal origin.
Dominant trait (gene) – predominant, manifesting - indicated in capital letters of the Latin alphabet: A, B, C, etc.
Recessive trait (gene) – the suppressed sign is indicated by the corresponding lowercase letter of the Latin alphabet: A,bWith etc
Analyzing crossing– crossing the test organism with another, which is a recessive homozygote for a given trait, which makes it possible to establish the genotype of the test person.
Dihybrid crossing– crossing of forms that differ from each other in two pairs of alternative characteristics.
Monohybrid crossing– crossing of forms that differ from each other in one pair of alternative characteristics.
Phenotype- the totality of all external signs and properties of an organism accessible to observation and analysis.
ü Algorithm for solving genetic problems
1. Read the task level carefully.
2. Make a brief note of the problem conditions.
3. Record the genotypes and phenotypes of the individuals being crossed.
4. Identify and record the types of gametes that are produced by the individuals being crossed.
5. Determine and record the genotypes and phenotypes of the offspring obtained from the cross.
6. Analyze the results of the crossing. To do this, determine the number of classes of offspring by phenotype and genotype and write them down as a numerical ratio.
7. Write down the answer to the problem question.
(When solving problems on certain topics, the sequence of stages may change and their content may be modified.)
ü Formatting tasks
1. It is customary to record the genotype of the female individual first, and then the male one ( correct entry - ♀ААВВ x ♂аавв; invalid entry - ♂aavv x ♀AABB).
2. Genes of one allelic pair are always written next to each other (correct entry - ♀ААВВ; incorrect entry ♀ААВВ).
3. When recording a genotype, letters denoting traits are always written in alphabetical order, regardless of which trait - dominant or recessive - they denote ( correct entry - ♀ааВВ; incorrect entry -♀ VVaa).
4. If only the phenotype of an individual is known, then when recording its genotype, only those genes are written whose presence is indisputable. A gene that cannot be determined by phenotype is designated with a “_”(for example, if the yellow color (A) and smooth shape (B) of pea seeds are dominant traits, and the green color (a) and wrinkled shape (c) are recessive, then the genotype of an individual with yellow wrinkled seeds is written as follows: A_vv).
5. The phenotype is always written under the genotype.
6. Gametes are written by circling them (A).
7. In individuals, the types of gametes are determined and recorded, not their number
correct entry incorrect entry
♀AA ♀AA
A A A
8. Phenotypes and types of gametes are written strictly under the corresponding genotype.
9. The progress of solving the problem is recorded with justification for each conclusion and the results obtained.
10. The results of crossing are always probabilistic nature and are expressed either as a percentage or as a fraction of a unit (for example, the probability of producing offspring susceptible to smut is 50%, or ½. The ratio of classes of offspring is written as a segregation formula (for example, yellow-seeded and green-seeded plants in a 1:1 ratio).
An example of solving and formatting problems
Task. In watermelon, green color (A) dominates over striped color. Determine the genotypes and phenotypes of F1 and F2 obtained from crossing homozygous plants with green and striped fruits.
The course uses geometric language, composed of notations and symbols adopted in a mathematics course (in particular, in the new geometry course in high school).
The whole variety of designations and symbols, as well as the connections between them, can be divided into two groups:
group I - designations of geometric figures and relationships between them;
group II designations of logical operations that form the syntactic basis of the geometric language.
Below is a complete list of math symbols used in this course. Particular attention is paid to the symbols that are used to indicate the projections of geometric figures.
Group I
SYMBOLS INDICATING GEOMETRIC FIGURES AND RELATIONS BETWEEN THEM
A. Designation of geometric figures
1. A geometric figure is designated - F.
2. Points are indicated by capital letters of the Latin alphabet or Arabic numerals:
A, B, C, D, ... , L, M, N, ...
1,2,3,4,...,12,13,14,...
3. Lines arbitrarily located in relation to the projection planes are designated by lowercase letters of the Latin alphabet:
a, b, c, d, ... , l, m, n, ...
Level lines are designated: h - horizontal; f- front.
The following notations are also used for straight lines:
(AB) - a straight line passing through points A and B;
[AB) - ray with beginning at point A;
[AB] - a straight line segment bounded by points A and B.
4. Surfaces are designated by lowercase letters of the Greek alphabet:
α, β, γ, δ,...,ζ,η,ν,...
To emphasize the way a surface is defined, the geometric elements by which it is defined should be indicated, for example:
α(a || b) - the plane α is determined by parallel lines a and b;
β(d 1 d 2 gα) - the surface β is determined by the guides d 1 and d 2, the generator g and the plane of parallelism α.
5. Angles are indicated:
∠ABC - angle with vertex at point B, as well as ∠α°, ∠β°, ... , ∠φ°, ...
6. Angular: the value (degree measure) is indicated by the sign, which is placed above the angle:
The magnitude of the angle ABC;
The magnitude of the angle φ.
A right angle is marked with a square with a dot inside
7. The distances between geometric figures are indicated by two vertical segments - ||.
For example:
|AB| - the distance between points A and B (length of segment AB);
|Aa| - distance from point A to line a;
|Aα| - distances from point A to surface α;
|ab| - distance between lines a and b;
|αβ| distance between surfaces α and β.
8. For projection planes, the following designations are accepted: π 1 and π 2, where π 1 is the horizontal projection plane;
π 2 - frontal projection plane.
When replacing projection planes or introducing new planes, the latter are designated π 3, π 4, etc.
9. The projection axes are designated: x, y, z, where x is the abscissa axis; y - ordinate axis; z - applicate axis.
Monge's constant straight line diagram is denoted by k.
10. Projections of points, lines, surfaces, any geometric figure are indicated by the same letters (or numbers) as the original, with the addition of a superscript corresponding to the projection plane on which they were obtained:
A", B", C", D", ... , L", M", N", horizontal projections of points; A", B", C", D", ... , L", M" , N", ... frontal projections of points; a" , b" , c" , d" , ... , l", m" , n" , - horizontal projections of lines; a" , b" , c" , d" , ... , l" , m " , n" , ... frontal projections of lines; α", β", γ", δ",...,ζ",η",ν",... horizontal projections of surfaces; α", β", γ", δ",...,ζ" ,η",ν",... frontal projections of surfaces.
11. Traces of planes (surfaces) are designated by the same letters as horizontal or frontal, with the addition of the subscript 0α, emphasizing that these lines lie in the projection plane and belong to the plane (surface) α.
So: h 0α - horizontal trace of the plane (surface) α;
f 0α - frontal trace of the plane (surface) α.
12. Traces of straight lines (lines) are indicated by capital letters, with which the words begin that define the name (in Latin transcription) of the projection plane that the line intersects, with a subscript indicating the affiliation with the line.
For example: H a - horizontal trace of a straight line (line) a;
F a - frontal trace of straight line (line) a.
13. The sequence of points, lines (any figure) is marked with subscripts 1,2,3,..., n:
A 1, A 2, A 3,..., A n;
a 1 , a 2 , a 3 ,...,a n ;
α 1, α 2, α 3,...,α n;
Ф 1, Ф 2, Ф 3,..., Ф n, etc.
The auxiliary projection of a point, obtained as a result of transformation to obtain the actual value of a geometric figure, is denoted by the same letter with a subscript 0:
A 0 , B 0 , C 0 , D 0 , ...
Axonometric projections
14. Axonometric projections of points, lines, surfaces are denoted by the same letters as nature with the addition of a superscript 0:
A 0, B 0, C 0, D 0, ...
1 0 , 2 0 , 3 0 , 4 0 , ...
a 0 , b 0 , c 0 , d 0 , ...
α 0 , β 0 , γ 0 , δ 0 , ...
15. Secondary projections are indicated by adding a superscript 1:
A 1 0, B 1 0, C 1 0, D 1 0, ...
1 1 0 , 2 1 0 , 3 1 0 , 4 1 0 , ...
a 1 0 , b 1 0 , c 1 0 , d 1 0 , ...
α 1 0 , β 1 0 , γ 1 0 , δ 1 0 , ...
To make it easier to read the drawings in the textbook, several colors are used when designing the illustrative material, each of which has a certain semantic meaning: black lines (dots) indicate the original data; green color is used for lines of auxiliary graphic constructions; red lines (dots) show the results of constructions or those geometric elements to which special attention should be paid.
| No. by por. | Designation | Content | Example of symbolic notation |
|---|---|---|---|
| 1 | ≡ | Match | (AB)≡(CD) - a straight line passing through points A and B, coincides with the line passing through points C and D |
| 2 | ≅ | Congruent | ∠ABC≅∠MNK - angle ABC is congruent to angle MNK |
| 3 | ∼ | Similar | ΔАВС∼ΔMNK - triangles АВС and MNK are similar |
| 4 | || | Parallel | α||β - plane α is parallel to plane β |
| 5 | ⊥ | Perpendicular | a⊥b - straight lines a and b are perpendicular |
| 6 | Crossbreed | c d - straight lines c and d intersect | |
| 7 | Tangents | t l - line t is tangent to line l. βα - plane β tangent to surface α |
|
| 8 | → | Displayed | F 1 →F 2 - figure F 1 is mapped to figure F 2 |
| 9 | S | Projection Center. If the projection center is an improper point, then its position is indicated by an arrow, indicating the direction of projection | - |
| 10 | s | Projection direction | - |
| 11 | P | Parallel projection | р s α Parallel projection - parallel projection onto the α plane in the s direction |
| No. by por. | Designation | Content | Example of symbolic notation | Example of symbolic notation in geometry |
|---|---|---|---|---|
| 1 | M,N | Sets | - | - |
| 2 | A,B,C,... | Elements of the set | - | - |
| 3 | { ... } | Comprises... | Ф(A, B, C,...) | Ф(A, B, C,...) - figure Ф consists of points A, B, C, ... |
| 4 | ∅ | Empty set | L - ∅ - the set L is empty (does not contain elements) | - |
| 5 | ∈ | Belongs to, is an element | 2∈N (where N is the set of natural numbers) - the number 2 belongs to the set N | A ∈ a - point A belongs to line a (point A lies on line a) |
| 6 | ⊂ | Includes, contains | N⊂M - set N is part (subset) of set M of all rational numbers | a⊂α - straight line a belongs to the plane α (understood in the sense: the set of points of the line a is a subset of the points of the plane α) |
| 7 | ∪ | An association | C = A U B - set C is a union of sets A and B; (1, 2. 3, 4.5) = (1,2,3)∪(4.5) | ABCD = ∪ [ВС] ∪ - broken line, ABCD is combining segments [AB], [BC], |
| 8 | ∩ | Intersection of many | M=K∩L - the set M is the intersection of the sets K and L (contains elements belonging to both the set K and the set L). M ∩ N = ∅ - the intersection of the sets M and N is the empty set (sets M and N do not have common elements) | a = α ∩ β - straight line a is the intersection planes α and β a ∩ b = ∅ - straight lines a and b do not intersect (do not have common points) |
| No. by por. | Designation | Content | Example of symbolic notation |
|---|---|---|---|
| 1 | ∧ | Conjunction of sentences; corresponds to the conjunction "and". A sentence (p∧q) is true if and only if p and q are both true | α∩β = (К:K∈α∧K∈β) The intersection of surfaces α and β is a set of points (line), consisting of all those and only those points K that belong to both the surface α and the surface β |
| 2 | ∨ | Disjunction of sentences; matches the conjunction "or". Sentence (p∨q) true when at least one of the sentences p or q is true (that is, either p or q, or both). | - |
| 3 | ⇒ | Implication is a logical consequence. The sentence p⇒q means: “if p, then q” | (a||c∧b||c)⇒a||b. If two lines are parallel to a third, then they are parallel to each other |
| 4 | ⇔ | The sentence (p⇔q) is understood in the sense: “if p, then also q; if q, then also p” | А∈α⇔А∈l⊂α. A point belongs to a plane if it belongs to some line belonging to this plane. The converse statement is also true: if a point belongs to a certain line, belonging to the plane, then it belongs to the plane itself |
| 5 | ∀ | The general quantifier reads: for everyone, for everyone, for anyone. The expression ∀(x)P(x) means: “for every x: the property P(x) holds” | ∀(ΔАВС)( = 180°) For any (for any) triangle, the sum of the values of its angles at vertices equals 180° |
| 6 | ∃ | The existential quantifier reads: exists. The expression ∃(x)P(x) means: “there is an x that has the property P(x)” | (∀α)(∃a).For any plane α there is a straight line a that does not belong to the plane α and parallel to the plane α |
| 7 | ∃1 | The quantifier of the uniqueness of existence, reads: there is only one (-i, -th)... The expression ∃1(x)(Рх) means: “there is only one (only one) x, having the property Px" | (∀ A, B)(A≠B)(∃1a)(a∋A, B) For any two different points A and B, there is a unique straight line a, passing through these points. |
| 8 | (Px) | Negation of the statement P(x) | ab(∃α)(α⊃a, b).If lines a and b intersect, then there is no plane a that contains them |
| 9 | \ | Negation of the sign | ≠ -segment [AB] is not equal to segment .a?b - line a is not parallel to line b |
Genetic symbolism
Symbolism is a list and explanation of conventional names and terms used in any branch of science.
The foundations of genetic symbolism were laid by Gregor Mendel, who used alphabetic symbolism to designate traits. Dominant traits were designated in capital letters of the Latin alphabet A, B, C, etc., recessive characters - in small letters - a, b, c, etc. Literal symbolism, proposed by Mendel, is essentially an algebraic form of expressing the laws of inheritance of characteristics.
The following symbolism is used to indicate crossing.
Parents are designated by the Latin letter P (Parents - parents), then their genotypes are written down next to them. The female gender is designated by the symbol ♂ (mirror of Venus), the male gender by ♀ (shield and spear of Mars). An “x” is placed between the parents to indicate crossing. The female genotype is written in first place, and the male in second.
The first generation is designated F 1 (Filli - children), second generation - F 2 etc. Nearby are the designations of the genotypes of the descendants.
Glossary of basic terms and concepts
Alleles (allelic genes)- different forms of one gene, resulting from mutations and located at identical points (loci) of paired homologous chromosomes.
Alternative signs– mutually exclusive, contrasting features.
Gametes (from the Greek “gametes” "- spouse) is a reproductive cell of a plant or animal organism that carries one gene from an allelic pair. Gametes always carry genes in a “pure” form, because are formed by meiotic cell division and contain one of a pair of homologous chromosomes.
Gen (from the Greek “genos” "- birth) is a section of a DNA molecule that carries information about the primary structure of one specific protein.
Allelic genes – paired genes located in identical regions of homologous chromosomes.
Genotype - a set of hereditary inclinations (genes) of an organism.
Heterozygote (from the Greek “heteros” " - other and zygote) - a zygote that has two different alleles for a given gene ( Aa, Bb).
Heterozygousare individuals that have received different genes from their parents. A heterozygous individual in its offspring produces segregation for this trait.
Homozygote (from the Greek "homos" " - identical and zygote) - a zygote that has the same alleles of a given gene (both dominant or both recessive).
Homozygous are called individuals who have received from their parents the same hereditary inclinations (genes) for some specific trait. A homozygous individual does not produce cleavage in its offspring.
Homologous chromosomes(from Greek “homos” " - identical) - paired chromosomes, identical in shape, size, set of genes. In a diploid cell, the set of chromosomes is always paired: one chromosome is from a pair of maternal origin, the second is of paternal origin.
Heterozygousare individuals that have received different genes from their parents. Thus, by genotype, individuals can be homozygous (AA or aa) or heterozygous (Aa).
Dominant trait (gene) – predominant, manifesting - indicated in capital letters of the Latin alphabet: A, B, C, etc.
Recessive trait (gene) – the suppressed sign is indicated by the corresponding lowercase letter of the Latin alphabet: a, b c, etc.
Analyzing crossing– crossing the test organism with another, which is a recessive homozygote for a given trait, which makes it possible to establish the genotype of the test person.
Dihybrid crossing– crossing of forms that differ from each other in two pairs of alternative characteristics.
Monohybrid crossing– crossing of forms that differ from each other in one pair of alternative characteristics.
Clean lines - organisms that are homozygous for one or more traits and do not produce manifestations of an alternative trait in their offspring.
Hairdryer is a sign.
Phenotype - the totality of all external signs and properties of an organism accessible to observation and analysis.
Algorithm for solving genetic problems
- Read the task level carefully.
- Make a brief note of the problem conditions.
- Record the genotypes and phenotypes of the individuals crossed.
- Identify and record the types of gametes that are produced by the individuals being crossed.
- Determine and record the genotypes and phenotypes of the offspring obtained from the cross.
- Analyze the results of the crossing. To do this, determine the number of classes of offspring by phenotype and genotype and write them down as a numerical ratio.
- Write down the answer to the question in the problem.
(When solving problems on certain topics, the sequence of stages may change and their content may be modified.)
Formatting tasks
- It is customary to record the female genotype first, and then the male (correct entry - ♀ААВВ x ♂аавв; invalid entry- ♂ aavv x ♀AABB).
- Genes of one allelic pair are always written next to each other(correct entry - ♀ААВВ; incorrect entry ♀ААВВ).
- When recording a genotype, letters denoting traits are always written in alphabetical order, regardless of which trait - dominant or recessive - they denote (correct entry - ♀ааВВ;incorrect entry -♀ VVaa).
- If only the phenotype of an individual is known, then when recording its genotype, only those genes whose presence is indisputable are written down.A gene that cannot be determined by phenotype is designated with a “_”(for example, if the yellow color (A) and smooth shape (B) of pea seeds are dominant traits, and the green color (a) and wrinkled shape (c) are recessive, then the genotype of an individual with yellow wrinkled seeds is written as follows: A_vv).
- The phenotype is always written under the genotype.
- Gametes are written by circling them.(A).
- In individuals, the types of gametes are determined and recorded, not their number
Infinity.J. Wallis (1655).
First found in the treatise of the English mathematician John Valis "On Conic Sections".
The base of natural logarithms. L. Euler (1736).
Mathematical constant, transcendental number. This number is sometimes called non-feathered in honor of the Scottish scientist Napier, author of the work “Description of the Amazing Table of Logarithms” (1614). The constant first appears tacitly in an appendix to the English translation of Napier's above-mentioned work, published in 1618. The constant itself was first calculated by the Swiss mathematician Jacob Bernoulli while solving the problem of the limiting value of interest income.
2,71828182845904523...
The first known use of this constant, where it was denoted by the letter b, found in Leibniz's letters to Huygens, 1690-1691. Letter e Euler began using it in 1727, and the first publication with this letter was his work “Mechanics, or the Science of Motion, Explained Analytically” in 1736. Respectively, e usually called Euler number. Why was the letter chosen? e, exactly unknown. Perhaps this is due to the fact that the word begins with it exponential(“indicative”, “exponential”). Another assumption is that the letters a, b, c And d have already been used quite widely for other purposes, and e was the first "free" letter.
The ratio of the circumference to the diameter. W. Jones (1706), L. Euler (1736).
Mathematical constant, irrational number. The number "pi", the old name is Ludolph's number. Like any irrational number, π is represented as an infinite non-periodic decimal fraction:
π =3.141592653589793...
For the first time, the designation of this number by the Greek letter π was used by the British mathematician William Jones in the book “A New Introduction to Mathematics”, and it became generally accepted after the work of Leonhard Euler. This designation comes from the initial letter of the Greek words περιφερεια - circle, periphery and περιμετρος - perimeter. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrienne Marie Legendre proved the irrationality of π 2 in 1774. Legendre and Euler assumed that π could be transcendental, i.e. cannot satisfy any algebraic equation with integer coefficients, which was eventually proven in 1882 by Ferdinand von Lindemann.
Imaginary unit. L. Euler (1777, in print - 1794).
It is known that the equation x 2 =1 has two roots: 1 And -1 . The imaginary unit is one of the two roots of the equation x 2 = -1, denoted by a Latin letter i, another root: -i. This designation was proposed by Leonhard Euler, who took the first letter of the Latin word for this purpose imaginarius(imaginary). He also extended all standard functions to the complex domain, i.e. set of numbers representable as a+ib, Where a And b- real numbers. The term "complex number" was introduced into widespread use by the German mathematician Carl Gauss in 1831, although the term had previously been used in the same sense by the French mathematician Lazare Carnot in 1803.
Unit vectors. W. Hamilton (1853).
Unit vectors are often associated with the coordinate axes of a coordinate system (in particular, the axes of a Cartesian coordinate system). Unit vector directed along the axis X, denoted i, unit vector directed along the axis Y, denoted j, and the unit vector directed along the axis Z, denoted k. Vectors i, j, k are called unit vectors, they have unit modules. The term "ort" was introduced by the English mathematician and engineer Oliver Heaviside (1892), and the notation i, j, k- Irish mathematician William Hamilton.
Integer part of the number, antie. K.Gauss (1808).
The integer part of the number [x] of the number x is the largest integer not exceeding x. So, =5, [-3,6]=-4. The function [x] is also called "antier of x". The whole-part function symbol was introduced by Carl Gauss in 1808. Some mathematicians prefer to use instead the notation E(x), proposed in 1798 by Legendre.
Angle of parallelism. N.I. Lobachevsky (1835).
On the Lobachevsky plane - the angle between the straight lineb, passing through the pointABOUTparallel to the linea, not containing a pointABOUT, and perpendicular fromABOUT on a. α - the length of this perpendicular. As the point moves awayABOUT from the straight line athe angle of parallelism decreases from 90° to 0°. Lobachevsky gave a formula for the angle of parallelismP( α )=2arctg e - α /q , Where q— some constant associated with the curvature of Lobachevsky space.
Unknown or variable quantities. R. Descartes (1637).
In mathematics, a variable is a quantity characterized by the set of values it can take. This may mean both a real physical quantity, temporarily considered in isolation from its physical context, and some abstract quantity that has no analogues in the real world. The concept of a variable arose in the 17th century. initially under the influence of the demands of natural science, which brought to the fore the study of movement, processes, and not just states. This concept required new forms for its expression. Such new forms were the letter algebra and analytical geometry of Rene Descartes. For the first time, the rectangular coordinate system and the notation x, y were introduced by Rene Descartes in his work “Discourse on Method” in 1637. Pierre Fermat also contributed to the development of the coordinate method, but his works were first published after his death. Descartes and Fermat used the coordinate method only on the plane. The coordinate method for three-dimensional space was first used by Leonhard Euler already in the 18th century.
Vector. O. Cauchy (1853).
From the very beginning, a vector is understood as an object that has a magnitude, a direction and (optionally) a point of application. The beginnings of vector calculus appeared along with the geometric model of complex numbers in Gauss (1831). Hamilton published developed operations with vectors as part of his quaternion calculus (the vector was formed by the imaginary components of the quaternion). Hamilton proposed the term vector(from the Latin word vector, carrier) and described some operations of vector analysis. Maxwell used this formalism in his works on electromagnetism, thereby drawing the attention of scientists to the new calculus. Soon Gibbs's Elements of Vector Analysis came out (1880s), and then Heaviside (1903) gave vector analysis its modern look. The vector sign itself was introduced into use by the French mathematician Augustin Louis Cauchy in 1853.
Addition, subtraction. J. Widman (1489).
The plus and minus signs were apparently invented in the German mathematical school of “Kossists” (that is, algebraists). They are used in Jan (Johannes) Widmann's textbook A Quick and Pleasant Account for All Merchants, published in 1489. Previously, addition was denoted by the letter p(from Latin plus"more") or Latin word et(conjunction “and”), and subtraction - letter m(from Latin minus"less, less") For Widmann, the plus symbol replaces not only addition, but also the conjunction “and.” The origin of these symbols is unclear, but most likely they were previously used in trading as indicators of profit and loss. Both symbols soon became common in Europe - with the exception of Italy, which continued to use the old designations for about a century.
Multiplication. W. Outred (1631), G. Leibniz (1698).
The multiplication sign in the form of an oblique cross was introduced in 1631 by the Englishman William Oughtred. Before him, the letter was most often used M, although other notations were also proposed: the rectangle symbol (French mathematician Erigon, 1634), asterisk (Swiss mathematician Johann Rahn, 1659). Later, Gottfried Wilhelm Leibniz replaced the cross with a dot (late 17th century) so as not to confuse it with the letter x; before him, such symbolism was found among the German astronomer and mathematician Regiomontanus (15th century) and the English scientist Thomas Herriot (1560 -1621).
Division. I.Ran (1659), G.Leibniz (1684).
William Oughtred used a slash / as a division sign. Gottfried Leibniz began to denote division with a colon. Before them, the letter was also often used D. Starting with Fibonacci, the horizontal line of the fraction is also used, which was used by Heron, Diophantus and in Arabic works. In England and the USA, the symbol ÷ (obelus), which was proposed by Johann Rahn (possibly with the participation of John Pell) in 1659, became widespread. An attempt by the American National Committee on Mathematical Standards ( National Committee on Mathematical Requirements) to remove obelus from practice (1923) was unsuccessful.
Percent. M. de la Porte (1685).
A hundredth of a whole, taken as a unit. The word “percent” itself comes from the Latin “pro centum”, which means “per hundred”. In 1685, the book “Manual of Commercial Arithmetic” by Mathieu de la Porte was published in Paris. In one place they talked about percentages, which were then designated “cto” (short for cento). However, the typesetter mistook this "cto" for a fraction and printed "%". So, due to a typo, this sign came into use.
Degrees. R. Descartes (1637), I. Newton (1676).
The modern notation for the exponent was introduced by Rene Descartes in his “ Geometry"(1637), however, only for natural powers with exponents greater than 2. Later, Isaac Newton extended this form of notation to negative and fractional exponents (1676), the interpretation of which had already been proposed by this time: the Flemish mathematician and engineer Simon Stevin, the English mathematician John Wallis and French mathematician Albert Girard.
Arithmetic root n-th power of a real number A≥0, - non-negative number n-th degree of which is equal to A. The arithmetic root of the 2nd degree is called a square root and can be written without indicating the degree: √. An arithmetic root of the 3rd degree is called a cube root. Medieval mathematicians (for example, Cardano) denoted the square root with the symbol R x (from the Latin Radix, root). The modern notation was first used by the German mathematician Christoph Rudolf, from the Cossist school, in 1525. This symbol comes from the stylized first letter of the same word radix. At first there was no line above the radical expression; it was later introduced by Descartes (1637) for a different purpose (instead of parentheses), and this feature soon merged with the root sign. In the 16th century, the cube root was denoted as follows: R x .u.cu (from lat. Radix universalis cubica). Albert Girard (1629) began to use the familiar notation for a root of an arbitrary degree. This format was established thanks to Isaac Newton and Gottfried Leibniz.
Logarithm, decimal logarithm, natural logarithm. I. Kepler (1624), B. Cavalieri (1632), A. Prinsheim (1893).
The term "logarithm" belongs to the Scottish mathematician John Napier ( “Description of the amazing table of logarithms”, 1614); it arose from a combination of the Greek words λογος (word, relation) and αριθμος (number). J. Napier's logarithm is an auxiliary number for measuring the ratio of two numbers. The modern definition of logarithm was first given by the English mathematician William Gardiner (1742). By definition, the logarithm of a number b based on a (a ≠ 1, a > 0) - exponent m, to which the number should be raised a(called the logarithm base) to get b. Designated log a b. So, m = log a b, If a m = b.
The first tables of decimal logarithms were published in 1617 by Oxford mathematics professor Henry Briggs. Therefore, abroad, decimal logarithms are often called Briggs logarithms. The term “natural logarithm” was introduced by Pietro Mengoli (1659) and Nicholas Mercator (1668), although the London mathematics teacher John Spidell compiled a table of natural logarithms back in 1619.
Until the end of the 19th century, there was no generally accepted notation for the logarithm, the basis a indicated to the left and above the symbol log, then above it. Ultimately, mathematicians came to the conclusion that the most convenient place for the base is below the line, after the symbol log. The logarithm sign - the result of the abbreviation of the word "logarithm" - appears in various forms almost simultaneously with the appearance of the first tables of logarithms, e.g. Log- by I. Kepler (1624) and G. Briggs (1631), log- by B. Cavalieri (1632). Designation ln for the natural logarithm was introduced by the German mathematician Alfred Pringsheim (1893).
Sine, cosine, tangent, cotangent. W. Outred (mid-17th century), I. Bernoulli (18th century), L. Euler (1748, 1753).
The abbreviations for sine and cosine were introduced by William Oughtred in the mid-17th century. Abbreviations for tangent and cotangent: tg, ctg introduced by Johann Bernoulli in the 18th century, they became widespread in Germany and Russia. In other countries the names of these functions are used tan, cot proposed by Albert Girard even earlier, at the beginning of the 17th century. Leonhard Euler (1748, 1753) brought the theory of trigonometric functions into its modern form, and we owe it to him for the consolidation of real symbolism.The term "trigonometric functions" was introduced by the German mathematician and physicist Georg Simon Klügel in 1770.
Indian mathematicians originally called the sine line "arha-jiva"(“half-string”, that is, half a chord), then the word "archa" was discarded and the sine line began to be called simply "jiva". Arabic translators did not translate the word "jiva" Arabic word "vatar", denoting string and chord, and transcribed in Arabic letters and began to call the sine line "jiba". Since in Arabic short vowels are not marked, but long “i” in the word "jiba" denoted in the same way as the semivowel “th”, the Arabs began to pronounce the name of the sine line "jibe", which literally means “hollow”, “sinus”. When translating Arabic works into Latin, European translators translated the word "jibe" Latin word sinus, having the same meaning.The term "tangent" (from lat.tangents- touching) was introduced by the Danish mathematician Thomas Fincke in his book The Geometry of the Round (1583).
Arcsine. K. Scherfer (1772), J. Lagrange (1772).
Inverse trigonometric functions are mathematical functions that are the inverse of trigonometric functions. The name of the inverse trigonometric function is formed from the name of the corresponding trigonometric function by adding the prefix "arc" (from Lat. arc- arc).The inverse trigonometric functions usually include six functions: arcsine (arcsin), arccosine (arccos), arctangent (arctg), arccotangent (arcctg), arcsecant (arcsec) and arccosecant (arccosec). Special symbols for inverse trigonometric functions were first used by Daniel Bernoulli (1729, 1736).Manner of denoting inverse trigonometric functions using a prefix arc(from lat. arcus, arc) appeared with the Austrian mathematician Karl Scherfer and was consolidated thanks to the French mathematician, astronomer and mechanic Joseph Louis Lagrange. It was meant that, for example, an ordinary sine allows one to find a chord subtending it along an arc of a circle, and the inverse function solves the opposite problem. Until the end of the 19th century, the English and German mathematical schools proposed other notations: sin -1 and 1/sin, but they are not widely used.
Hyperbolic sine, hyperbolic cosine. V. Riccati (1757).
Historians discovered the first appearance of hyperbolic functions in the works of the English mathematician Abraham de Moivre (1707, 1722). A modern definition and a detailed study of them was carried out by the Italian Vincenzo Riccati in 1757 in his work “Opusculorum”, he also proposed their designations: sh,ch. Riccati started from considering the unit hyperbola. An independent discovery and further study of the properties of hyperbolic functions was carried out by the German mathematician, physicist and philosopher Johann Lambert (1768), who established the wide parallelism of the formulas of ordinary and hyperbolic trigonometry. N.I. Lobachevsky subsequently used this parallelism in an attempt to prove the consistency of non-Euclidean geometry, in which ordinary trigonometry is replaced by hyperbolic one.
Just as the trigonometric sine and cosine are the coordinates of a point on the coordinate circle, the hyperbolic sine and cosine are the coordinates of a point on a hyperbola. Hyperbolic functions are expressed in terms of an exponential and are closely related to trigonometric functions: sh(x)=0.5(e x -e -x) , ch(x)=0.5(e x +e -x). By analogy with trigonometric functions, hyperbolic tangent and cotangent are defined as the ratios of hyperbolic sine and cosine, cosine and sine, respectively.
Differential. G. Leibniz (1675, published 1684).
The main, linear part of the function increment.If the function y=f(x) one variable x has at x=x 0derivative, and incrementΔy=f(x 0 +?x)-f(x 0)functions f(x) can be represented in the formΔy=f"(x 0 )Δx+R(Δx) , where is the member R infinitesimal compared toΔx. First memberdy=f"(x 0 )Δxin this expansion and is called the differential of the function f(x) at the pointx 0. IN works of Gottfried Leibniz, Jacob and Johann Bernoulli the word"differentia"was used in the sense of “increment”, it was denoted by I. Bernoulli through Δ. G. Leibniz (1675, published 1684) used the notation for the “infinitesimal difference”d- the first letter of the word"differential", formed by him from"differentia".
Indefinite integral. G. Leibniz (1675, published 1686).
The word "integral" was first used in print by Jacob Bernoulli (1690). Perhaps the term is derived from the Latin integer- whole. According to another assumption, the basis was the Latin word integro- bring to its previous state, restore. The sign ∫ is used to represent an integral in mathematics and is a stylized representation of the first letter of the Latin word summa - sum. It was first used by the German mathematician and founder of differential and integral calculus, Gottfried Leibniz, at the end of the 17th century. Another of the founders of differential and integral calculus, Isaac Newton, did not propose an alternative symbolism for the integral in his works, although he tried various options: a vertical bar above the function or a square symbol that stands in front of the function or borders it. Indefinite integral for a function y=f(x) is the set of all antiderivatives of a given function.
Definite integral. J. Fourier (1819-1822).
Definite integral of a function f(x) with a lower limit a and upper limit b can be defined as the difference F(b) - F(a) = a ∫ b f(x)dx , Where F(x)- some antiderivative of a function f(x) . Definite integral a ∫ b f(x)dx numerically equal to the area of the figure bounded by the x-axis and straight lines x=a And x=b and the graph of the function f(x). The design of a definite integral in the form we are familiar with was proposed by the French mathematician and physicist Jean Baptiste Joseph Fourier at the beginning of the 19th century.
Derivative. G. Leibniz (1675), J. Lagrange (1770, 1779).
Derivative is the basic concept of differential calculus, characterizing the rate of change of a function f(x) when the argument changes x . It is defined as the limit of the ratio of the increment of a function to the increment of its argument as the increment of the argument tends to zero, if such a limit exists. A function that has a finite derivative at some point is called differentiable at that point. The process of calculating the derivative is called differentiation. The reverse process is integration. In classical differential calculus, the derivative is most often defined through the concepts of the theory of limits, but historically the theory of limits appeared later than differential calculus.
The term “derivative” was introduced by Joseph Louis Lagrange in 1797, the denotation of a derivative using a stroke is also used by him (1770, 1779), and dy/dx- Gottfried Leibniz in 1675. The manner of denoting the time derivative with a dot over a letter comes from Newton (1691).The Russian term “derivative of a function” was first used by a Russian mathematicianVasily Ivanovich Viskovatov (1779-1812).
Partial derivative. A. Legendre (1786), J. Lagrange (1797, 1801).
For functions of many variables, partial derivatives are defined - derivatives with respect to one of the arguments, calculated under the assumption that the remaining arguments are constant. Designations ∂f/ ∂ x, ∂ z/ ∂ y introduced by French mathematician Adrien Marie Legendre in 1786; fx",z x "- Joseph Louis Lagrange (1797, 1801); ∂ 2 z/ ∂ x 2, ∂ 2 z/ ∂ x ∂ y- partial derivatives of the second order - German mathematician Carl Gustav Jacob Jacobi (1837).
Difference, increment. I. Bernoulli (late 17th century - first half of the 18th century), L. Euler (1755).
The designation of increment by the letter Δ was first used by the Swiss mathematician Johann Bernoulli. The delta symbol came into general use after the work of Leonhard Euler in 1755.
Sum. L. Euler (1755).
Sum is the result of adding quantities (numbers, functions, vectors, matrices, etc.). To denote the sum of n numbers a 1, a 2, ..., a n, the Greek letter “sigma” Σ is used: a 1 + a 2 + ... + a n = Σ n i=1 a i = Σ n 1 a i. The Σ sign for the sum was introduced by Leonhard Euler in 1755.
Work. K.Gauss (1812).
A product is the result of multiplication. To denote the product of n numbers a 1, a 2, ..., a n, the Greek letter pi Π is used: a 1 · a 2 · ... · a n = Π n i=1 a i = Π n 1 a i. For example, 1 · 3 · 5 · ... · 97 · 99 = ? 50 1 (2i-1). The Π sign for a product was introduced by the German mathematician Carl Gauss in 1812. In Russian mathematical literature, the term “product” was first encountered by Leonty Filippovich Magnitsky in 1703.
Factorial. K. Crump (1808).
The factorial of a number n (denoted n!, pronounced "en factorial") is the product of all natural numbers up to n inclusive: n! = 1·2·3·...·n. For example, 5! = 1·2·3·4·5 = 120. By definition, 0 is assumed! = 1. Factorial is defined only for non-negative integers. The factorial of n is equal to the number of permutations of n elements. For example, 3! = 6, indeed,
♣ ♦
♣ ♦
♣ ♦
♦ ♣
♦ ♣
♦ ♣
All six and only six permutations of three elements.
The term "factorial" was introduced by the French mathematician and politician Louis Francois Antoine Arbogast (1800), the designation n! - French mathematician Christian Crump (1808).
Modulus, absolute value. K. Weierstrass (1841).
The absolute value of a real number x is a non-negative number defined as follows: |x| = x for x ≥ 0, and |x| = -x for x ≤ 0. For example, |7| = 7, |- 0.23| = -(-0.23) = 0.23. The modulus of a complex number z = a + ib is a real number equal to √(a 2 + b 2).
It is believed that the term “module” was proposed by the English mathematician and philosopher, Newton’s student, Roger Cotes. Gottfried Leibniz also used this function, which he called “modulus” and denoted: mol x. The generally accepted notation for absolute value was introduced in 1841 by the German mathematician Karl Weierstrass. For complex numbers, this concept was introduced by French mathematicians Augustin Cauchy and Jean Robert Argan at the beginning of the 19th century. In 1903, the Austrian scientist Konrad Lorenz used the same symbolism for the length of a vector.
Norm. E. Schmidt (1908).
A norm is a functional defined on a vector space and generalizing the concept of the length of a vector or modulus of a number. The "norm" sign (from the Latin word "norma" - "rule", "pattern") was introduced by the German mathematician Erhard Schmidt in 1908.
Limit. S. Lhuillier (1786), W. Hamilton (1853), many mathematicians (until the beginning of the twentieth century)
Limit is one of the basic concepts of mathematical analysis, meaning that a certain variable value in the process of its change under consideration indefinitely approaches a certain constant value. The concept of a limit was used intuitively in the second half of the 17th century by Isaac Newton, as well as by 18th-century mathematicians such as Leonhard Euler and Joseph Louis Lagrange. The first rigorous definitions of the sequence limit were given by Bernard Bolzano in 1816 and Augustin Cauchy in 1821. The symbol lim (the first 3 letters from the Latin word limes - border) appeared in 1787 by the Swiss mathematician Simon Antoine Jean Lhuillier, but its use did not yet resemble modern ones. The expression lim in a more familiar form was first used by the Irish mathematician William Hamilton in 1853.Weierstrass introduced a designation close to the modern one, but instead of the familiar arrow, he used an equal sign. The arrow appeared at the beginning of the 20th century among several mathematicians at once - for example, the English mathematician Godfried Hardy in 1908.
Zeta function, d Riemann zeta function. B. Riemann (1857).
Analytical function of a complex variable s = σ + it, for σ > 1, determined absolutely and uniformly by a convergent Dirichlet series:
ζ(s) = 1 -s + 2 -s + 3 -s + ... .
For σ > 1, the representation in the form of the Euler product is valid:
ζ(s) = Π p (1-p -s) -s,
where the product is taken over all prime p. The zeta function plays a big role in number theory.As a function of a real variable, the zeta function was introduced in 1737 (published in 1744) by L. Euler, who indicated its expansion into a product. This function was then considered by the German mathematician L. Dirichlet and, especially successfully, by the Russian mathematician and mechanic P.L. Chebyshev when studying the law of distribution of prime numbers. However, the most profound properties of the zeta function were discovered later, after the work of the German mathematician Georg Friedrich Bernhard Riemann (1859), where the zeta function was considered as a function of a complex variable; He also introduced the name “zeta function” and the designation ζ(s) in 1857.
Gamma function, Euler Γ function. A. Legendre (1814).
The Gamma function is a mathematical function that extends the concept of factorial to the field of complex numbers. Usually denoted by Γ(z). The G-function was first introduced by Leonhard Euler in 1729; it is determined by the formula:
Γ(z) = limn→∞ n!·n z /z(z+1)...(z+n).
A large number of integrals, infinite products and sums of series are expressed through the G-function. Widely used in analytical number theory. The name "Gamma function" and the notation Γ(z) were proposed by the French mathematician Adrien Marie Legendre in 1814.
Beta function, B function, Euler B function. J. Binet (1839).
A function of two variables p and q, defined for p>0, q>0 by the equality:
B(p, q) = 0 ∫ 1 x p-1 (1-x) q-1 dx.
The beta function can be expressed through the Γ-function: B(p, q) = Γ(p)Г(q)/Г(p+q).Just as the gamma function for integers is a generalization of factorial, the beta function is, in a sense, a generalization of binomial coefficients.
The beta function describes many propertieselementary particles participating in strong interaction. This feature was noticed by the Italian theoretical physicistGabriele Veneziano in 1968. This marked the beginning string theory.
The name "beta function" and the designation B(p, q) were introduced in 1839 by the French mathematician, mechanic and astronomer Jacques Philippe Marie Binet.
Laplace operator, Laplacian. R. Murphy (1833).
Linear differential operator Δ, which assigns functions φ(x 1, x 2, ..., x n) of n variables x 1, x 2, ..., x n:
Δφ = ∂ 2 φ/∂х 1 2 + ∂ 2 φ/∂х 2 2 + ... + ∂ 2 φ/∂х n 2.
In particular, for a function φ(x) of one variable, the Laplace operator coincides with the operator of the 2nd derivative: Δφ = d 2 φ/dx 2 . The equation Δφ = 0 is usually called Laplace's equation; This is where the names “Laplace operator” or “Laplacian” come from. The designation Δ was introduced by the English physicist and mathematician Robert Murphy in 1833.
Hamilton operator, nabla operator, Hamiltonian. O. Heaviside (1892).
Vector differential operator of the form
∇ = ∂/∂x i+ ∂/∂y · j+ ∂/∂z · k,
Where i, j, And k- coordinate unit vectors. The basic operations of vector analysis, as well as the Laplace operator, are expressed in a natural way through the Nabla operator.
In 1853, Irish mathematician William Rowan Hamilton introduced this operator and coined the symbol ∇ for it as an inverted Greek letter Δ (delta). In Hamilton, the tip of the symbol pointed to the left; later, in the works of the Scottish mathematician and physicist Peter Guthrie Tate, the symbol acquired its modern form. Hamilton called this symbol "atled" (the word "delta" read backwards). Later, English scholars, including Oliver Heaviside, began to call this symbol "nabla", after the name of the letter ∇ in the Phoenician alphabet, where it occurs. The origin of the letter is associated with a musical instrument such as the harp, ναβλα (nabla) in ancient Greek meaning “harp”. The operator was called the Hamilton operator, or nabla operator.
Function. I. Bernoulli (1718), L. Euler (1734).
A mathematical concept that reflects the relationship between elements of sets. We can say that a function is a “law”, a “rule” according to which each element of one set (called the domain of definition) is associated with some element of another set (called the domain of values). The mathematical concept of a function expresses the intuitive idea of how one quantity completely determines the value of another quantity. Often the term "function" refers to a numerical function; that is, a function that puts some numbers in correspondence with others. For a long time, mathematicians specified arguments without parentheses, for example, like this - φх. This notation was first used by the Swiss mathematician Johann Bernoulli in 1718.Parentheses were used only in the case of multiple arguments or if the argument was a complex expression. Echoes of those times are the recordings still in use todaysin x, log xetc. But gradually the use of parentheses, f(x) , became a general rule. And the main credit for this belongs to Leonhard Euler.
Equality. R. Record (1557).
The equals sign was proposed by the Welsh physician and mathematician Robert Record in 1557; the outline of the symbol was much longer than the current one, as it imitated the image of two parallel segments. The author explained that there is nothing more equal in the world than two parallel segments of the same length. Before this, in ancient and medieval mathematics equality was denoted verbally (for example est egale). In the 17th century, Rene Descartes began to use æ (from lat. aequalis), and he used the modern equal sign to indicate that the coefficient can be negative. François Viète used the equal sign to denote subtraction. The Record symbol did not become widespread immediately. The spread of the Record symbol was hampered by the fact that since ancient times the same symbol was used to indicate the parallelism of straight lines; In the end, it was decided to make the parallelism symbol vertical. In continental Europe, the "=" sign was introduced by Gottfried Leibniz only at the turn of the 17th-18th centuries, that is, more than 100 years after the death of Robert Record, who first used it for this purpose.
Approximately equal, approximately equal. A.Gunther (1882).
Sign " ≈ " was introduced into use as a symbol for the relation "approximately equal" by the German mathematician and physicist Adam Wilhelm Sigmund Günther in 1882.
More less. T. Harriot (1631).
These two signs were introduced into use by the English astronomer, mathematician, ethnographer and translator Thomas Harriot in 1631; before that, the words “more” and “less” were used.
Comparability. K.Gauss (1801).
Comparison is a relationship between two integers n and m, meaning that the difference n-m of these numbers is divided by a given integer a, called the comparison modulus; it is written: n≡m(mod а) and reads “the numbers n and m are comparable modulo a”. For example, 3≡11(mod 4), since 3-11 is divisible by 4; the numbers 3 and 11 are comparable modulo 4. Congruences have many properties similar to those of equalities. Thus, a term located in one part of the comparison can be transferred with the opposite sign to another part, and comparisons with the same module can be added, subtracted, multiplied, both parts of the comparison can be multiplied by the same number, etc. For example,
3≡9+2(mod 4) and 3-2≡9(mod 4)
At the same time true comparisons. And from a pair of correct comparisons 3≡11(mod 4) and 1≡5(mod 4) the following follows:
3+1≡11+5(mod 4)
3-1≡11-5(mod 4)
3·1≡11·5(mod 4)
3 2 ≡11 2 (mod 4)
3·23≡11·23(mod 4)
In number theory, methods for solving various comparisons are considered, i.e. methods for finding integers that satisfy comparisons of one type or another. Modulo comparisons were first used by the German mathematician Carl Gauss in his 1801 book Arithmetic Studies. He also proposed symbolism for comparisons that was established in mathematics.
Identity. B. Riemann (1857).
Identity is the equality of two analytical expressions, valid for any permissible values of the letters included in it. The equality a+b = b+a is valid for all numerical values of a and b, and therefore is an identity. To record identities, in some cases, since 1857, the sign “≡” (read “identically equal”) has been used, the author of which in this use is the German mathematician Georg Friedrich Bernhard Riemann. You can write down a+b ≡ b+a.
Perpendicularity. P. Erigon (1634).
Perpendicularity is the relative position of two straight lines, planes, or a straight line and a plane, in which the indicated figures form a right angle. The sign ⊥ to denote perpendicularity was introduced in 1634 by the French mathematician and astronomer Pierre Erigon. The concept of perpendicularity has a number of generalizations, but all of them, as a rule, are accompanied by the sign ⊥.
Parallelism. W. Outred (posthumous edition 1677).
Parallelism is the relationship between certain geometric figures; for example, straight. Defined differently depending on different geometries; for example, in the geometry of Euclid and in the geometry of Lobachevsky. The sign of parallelism has been known since ancient times, it was used by Heron and Pappus of Alexandria. At first, the symbol was similar to the current equals sign (only more extended), but with the advent of the latter, to avoid confusion, the symbol was turned vertically ||. It appeared in this form for the first time in the posthumous edition of the works of the English mathematician William Oughtred in 1677.
Intersection, union. J. Peano (1888).
The intersection of sets is a set that contains those and only those elements that simultaneously belong to all given sets. A union of sets is a set that contains all the elements of the original sets. Intersection and union are also called operations on sets that assign new sets to certain ones according to the rules indicated above. Denoted by ∩ and ∪, respectively. For example, if
A= (♠ ♣ ) And B= (♣ ♦),
That
A∩B= {♣ }
A∪B= {♠ ♣ ♦ } .
Contains, contains. E. Schroeder (1890).
If A and B are two sets and there are no elements in A that do not belong to B, then they say that A is contained in B. They write A⊂B or B⊃A (B contains A). For example,
{♠}⊂{♠ ♣}⊂{♠ ♣ ♦ }
{♠ ♣ ♦ }⊃{ ♦ }⊃{♦ }
The symbols “contains” and “contains” appeared in 1890 by the German mathematician and logician Ernst Schroeder.
Affiliation. J. Peano (1895).
If a is an element of the set A, then write a∈A and read “a belongs to A.” If a is not an element of the set A, write a∉A and read “a does not belong to A.” At first, the relations “contained” and “belongs” (“is an element”) were not distinguished, but over time these concepts required differentiation. The symbol ∈ was first used by the Italian mathematician Giuseppe Peano in 1895. The symbol ∈ comes from the first letter of the Greek word εστι - to be.
Quantifier of universality, quantifier of existence. G. Gentzen (1935), C. Pierce (1885).
Quantifier is a general name for logical operations that indicate the domain of truth of a predicate (mathematical statement). Philosophers have long paid attention to logical operations that limit the domain of truth of a predicate, but have not identified them as a separate class of operations. Although quantifier-logical constructions are widely used in both scientific and everyday speech, their formalization occurred only in 1879, in the book of the German logician, mathematician and philosopher Friedrich Ludwig Gottlob Frege “The Calculus of Concepts”. Frege's notation looked like cumbersome graphic constructions and was not accepted. Subsequently, many more successful symbols were proposed, but the notations that became generally accepted were ∃ for the existential quantifier (read “exists”, “there is”), proposed by the American philosopher, logician and mathematician Charles Peirce in 1885, and ∀ for the universal quantifier (read “any” , “each”, “everyone”), formed by the German mathematician and logician Gerhard Karl Erich Gentzen in 1935 by analogy with the symbol of the quantifier of existence (inverted first letters of the English words Existence (existence) and Any (any)). For example, record
(∀ε>0) (∃δ>0) (∀x≠x 0 , |x-x 0 |<δ) (|f(x)-A|<ε)
reads like this: “for any ε>0 there is δ>0 such that for all x not equal to x 0 and satisfying the inequality |x-x 0 |<δ, выполняется неравенство |f(x)-A|<ε".
Empty set. N. Bourbaki (1939).
A set that does not contain a single element. The sign of the empty set was introduced in the books of Nicolas Bourbaki in 1939. Bourbaki is the collective pseudonym of a group of French mathematicians created in 1935. One of the members of the Bourbaki group was Andre Weil, the author of the Ø symbol.
Q.E.D. D. Knuth (1978).
In mathematics, proof is understood as a sequence of reasoning built on certain rules, showing that a certain statement is true. Since the Renaissance, the end of a proof has been denoted by mathematicians by the abbreviation "Q.E.D.", from the Latin expression "Quod Erat Demonstrandum" - "What was required to be proved." When creating the computer layout system ΤΕΧ in 1978, American computer science professor Donald Edwin Knuth used a symbol: a filled square, the so-called “Halmos symbol”, named after the Hungarian-born American mathematician Paul Richard Halmos. Today, the completion of a proof is usually indicated by the Halmos Symbol. As an alternative, other signs are used: an empty square, a right triangle, // (two forward slashes), as well as the Russian abbreviation “ch.t.d.”
Heredity is the ability of organisms to transmit their characteristics and properties to the next generation, i.e. the ability to reproduce their own kind.
A gene is a section of a DNA molecule that carries information about the structure of one protein.
Genotype is the totality of all hereditary properties of an individual, the hereditary basis of an organism, made up of a set of genes.
Phenotype is the totality of all internal and external characteristics and properties of an individual, formed on the basis of the genotype in the process of its individual development.
Monohybrid crossing is the crossing of parental forms that differ hereditarily in only one pair of traits.
Dominance is the phenomenon of predominance of traits during crossing.
Dominant trait - predominant.
A recessive trait is one that recedes or disappears.
Homozygotes are individuals that, when self-pollinating for a given pair of traits, produce homogeneous, non-splitting offspring.
Heterozygotes are individuals that exhibit splitting according to a given pair of characteristics.
Alleles are different forms of the same gene.
Dihybrid crossing is the crossing of parental forms that differ in two pairs of characteristics.
Variability is the ability of organisms to change their characteristics and properties.
Modifying (phenotypic) variability - changes in the phenotype that occur under the influence of changes in external conditions and are not associated with changes in the genotype.
The reaction norm is the limits of modification variability of a given trait.
Mutations are changes in the genotype caused by structural changes in genes or chromosomes.
Polyploidy is an increase in chromosomes in a cell that is a multiple of the haploid number (3n, 4n or more).
In genetics, the following generally accepted symbols are used:
- the letter P (from the Latin “parenta” - parents) denotes the parent organisms taken for crossing;
- the sign ♀ (“mirror of Venus”) - denotes the female gender;
- ♂ (“shield and spear of Mars”) - denote a male iol.
- Crossing is designated by the sign “X”, the hybrid offspring is designated by the letter F (from the Latin “philia” - children) with a number corresponding to the serial number of the generation - F 1, F 2, F 3.
Laws formulated by G. Mendel
Dominance Rule, or the first law: during monohybrid crossing, only dominant traits appear in first-generation hybrids - it is phenotypically uniform.
Law of splitting, or the second law of G. Mendel: when crossing hybrids of the first generation, the characteristics in the offspring are split in a ratio of 3:1 - two phenotypic groups are formed - dominant and recessive.
Law of independent inheritance(third law): during dihybrid crossing in hybrids, each pair of traits is inherited independently of the others and gives different combinations with it. Four phenotypic groups are formed, characterized by a ratio of 9:3:3:1.
Progress of monohybrid crossing (Mendel's first and second laws)
Light circles - organisms with dominant traits; dark - with a recessive trait.
Gamete purity hypothesis: pairs of alternative characteristics found in each organism do not mix and during the formation of gametes, one from each pair passes into them in their pure form.
To explain the observed patterns, Mendel put forward the hypothesis of gamete purity, suggesting the following:
- any trait is formed under the influence of a material factor (gene).
- He defined the factor that determines a dominant trait with a capital letter A, and a recessive trait with a capital letter. Each individual contains two factors that determine the development of the trait, one it receives from the mother, the other from the father.
- During the formation of gametes in animals and spores - in plants, a reduction of factors occurs and only one enters each gamete or spore.
According to this hypothesis, the course of a monohybrid cross is written as follows:
For any combination of gametes, all hybrids have the same genotype and phenotype.
In F 2, the genotype split will be 1AA; 2Aa; 1aa, but to the phenotype: 3 yellow, 1 green (3:1).
Sometimes F1 hybrids do not have complete dominance; their characteristics are intermediate. This type of inheritance is called intermediate, or incomplete dominance.
Example: monohybrid crossing of a night beauty: with incomplete dominance in F2, splitting by phenotype and genotype is expressed by the same ratio: 1:2:1 (1 white, 2 pink, 1 red).
The nature of inheritance was defined as independent and Mendel's third law, or the law of independent inheritance, was formulated.
Independent inheritance is of great importance for evolution, as it is the source of combinative variability and diversity of living organisms.
Law of chained inheritance
In 1911, Thomas Morgan formulated law of chained inheritance- linked genes localized on the same chromosome are inherited together and do not show independent segregation.
Each chromosome contains several thousand genes that distinguish one individual of a given species from another. Clarifying the question of how the characteristics of these genes will be inherited, Morgan established that genes located on the same chromosome are inherited linked together, as one alternative pair, without revealing independent inheritance.
Cohesion is not always absolute. In the prophase of the first division of meiosis, during the conjugation of chromosomes, their crossover occurs, as a result of which genes located on one chromosome ended up on different homologous chromosomes and ended up in different gametes.
Chromosome crossing diagram
Two genes located on the same chromosome (open circles on one of the chromosomes) end up on different homologous chromosomes as a result of crossover.
Such an exchange leads to a rearrangement of linked genes and is one of the sources of combinative variability.
Chromosome crossing plays a role in evolution, as a new combination of genes causes the appearance of new traits that can be beneficial or harmful to the organism and affect its survival.
A gene can simultaneously influence the formation of several traits, while exhibiting multiple effects.
