The following functions are relative. Functions of economic relations
Let be r Н X X Y.
functional relation is a binary relation r, in which each element corresponds exactly one such that the pair belongs to a relation or such does not exist at all: or.
Functional relationship - this is a binary relation r, for which: .
Everywhere a certain relation– binary relation r, for which D r =X("there are no lonely X").
Surjective relation– binary relation r, for which J r = Y("there are no lonely y").
Injective relation is a binary relation in which different X correspond to different at.
Bijection– functional, everywhere defined, injective, surjective relation, defines a one-to-one correspondence of sets.
for example:
Let be r= ( (x, y) О R 2 | y 2 + x 2 = 1, y > 0 ).
Attitude r- functional,
not everywhere defined ("there are lonely X"),
not injective (there are different X, at),
not surjective ("there are lonely at"),
not a bijection.
For example:
Let K= ((x,y) н R 2 | y = x+1)
The relation à is functional,
The relation Ã- is defined everywhere ("there are no lonely X"),
The relation Ã- is injective (there are no different X, which correspond to the same at),
The relation Ã- is surjective ("there are no lonely at"),
The relation à is bijective, a mutually homogeneous correspondence.
For example:
Let j=((1,2), (2,3), (1,3), (3,4), (2,4), (1,4)) be defined on the set N 4.
The relation j is not functional, x=1 corresponds to three y: (1,2), (1,3), (1,4)
Relation j - not everywhere defined D j =(1,2,3)¹ N 4
Relation j - not surjective I j =(1,2,3)¹ N 4
The relation j is not injective, different x correspond to the same y, for example (2,3) and (1,3).
Assignment for laboratory work
1. Sets are given N1 and N2. Calculate sets:
(N1 X N2) З (N2 X N1);
(N1 X N2) È (N2 X N1);
(N1 Ç N2) x (N1 Ç N2);
(N1 → N2) x (N1 → N2),
where N1 = ( digits of the record book number, the last three };
N2 = ( date and month numbers of birth }.
2. Relationships r and g set on the set N 6 \u003d (1,2,3,4,5,6).
Describe relationships r,g,r -1 , r ○g, r- 1 ○g couples list.
Find relationship matrices r and g.
For each relation, define the domain of definition and range of values.
Define relationship properties.
Identify equivalence relations and construct equivalence classes.
Identify order relationships and classify them.
1) r= { (m,n) | m > n)
g= { (m,n) | comparison modulo 2 }
2) r= { (m,n) | (m - n) divisible by 2 }
g= { (m,n) | m divider n)
3) r= { (m,n) | m< n }
g= { (m,n) | comparison modulo 3 }
4) r= { (m,n) | (m+n)- even }
g= { (m,n) | m 2 \u003d n)
5) r= { (m,n) | m/n- degree 2 }
g= { (m,n) | m = n)
6) r= { (m,n) | m/n- even }
g = ((m,n) | m³ n)
7) r= { (m,n) | m/n- odd }
g= { (m,n) | comparison modulo 4 }
8) r= { (m,n) | m*n- even }
g= { (m,n) | m£ n)
9) r= { (m,n) | comparison modulo 5 }
g= { (m,n) | m divided by n)
10) r= { (m,n) | m- even, n- even }
g= { (m,n) | m divider n)
11) r= { (m,n) | m = n)
g= { (m,n) | (m+n)£ 5 }
12) r={ (m,n) | m and n have the same remainder when divided by 3 }
g= { (m,n) | (m-n)³2 }
13) r= { (m,n) | (m+n) is divisible by 2 }
g = ((m,n) | £2 (m-n)£4 }
14) r= { (m,n) | (m+n) divisible by 3 }
g= { (m,n) | m¹ n)
15) r= { (m,n) | m and n have a common divisor }
g= { (m,n) | m2£ n)
16) r= { (m,n) | (m - n) is divisible by 2 }
g= { (m,n) | m< n +2 }
17) r= { (m,n) | comparison modulo 4 }
g= { (m,n) | m£ n)
18) r= { (m,n) | m divided entirely into n)
g= { (m,n) | m¹ n, m- even }
19) r= { (m,n) | comparison modulo 3 }
g= { (m,n) | 1 £ (m-n)£3 }
20) r= { (m,n) | (m - n) divisible by 4 }
g= { (m,n) | m¹ n)
21) r= { (m,n) | m- odd, n- odd }
g= { (m,n) | m£ n, n- even }
22) r= { (m,n) | m and n have an odd remainder when divided by 3 }
g= { (m,n) | (m-n)³1 }
23) r= { (m,n) | m*n- odd }
g= { (m,n) | comparison modulo 2 }
24) r= { (m,n) | m*n- even }
g= { (m,n) | 1 £ (m-n)£3 }
25) r= { (m,n) | (m+ n)- even }
g= { (m,n) | m not divisible by n)
26) r= { (m,n) | m = n)
g= { (m,n) | m divided entirely into n)
27) r= { (m,n) | (m-n)- even }
g= { (m,n) | m divider n)
28) r= { (m,n) | (m-n)³2 }
g= { (m,n) | m divided entirely into n)
29) r= { (m,n) | m2³ n)
g= { (m,n) | m / n- odd }
30) r= { (m,n) | m³ n, m - even }
g= { (m,n) | m and n have a common divisor other than 1 }
3. Determine if the given relation is f- functional, everywhere defined, injective, surjective, bijection ( R is the set of real numbers). Build a graph of the relationship, determine the domain of definition and range of values.
Do the same task for relationships r and g from point 3 of the laboratory work.
1) f=( (x, y) Î R 2 | y=1/x +7x )
2) f=( (x, y) Î R 2 | x³ y )
3) f=( (x, y) Î R 2 | y³ x )
4) f=( (x, y) Î R 2 | y³ x, x³ 0 }
5) f=( (x, y) Î R 2 | y 2 + x 2 = 1)
6) f=( (x, y) Î R 2 | 2 | y | + | x | = 1 )
7) f=( (x, y) Î R 2 | x+y£ 1 }
8) f=( (x, y) Î R 2 | x = y 2 )
9) f=( (x, y) Î R 2 | y = x 3 + 1)
10) f=( (x, y) Î R 2 | y = -x 2 )
11) f=( (x, y) Î R 2 | | y | + | x | = 1 )
12) f=( (x, y) Î R 2 | x = y -2 )
13) f=( (x, y) Î R 2 | y2 + x2³ 1,y> 0 }
14) f=( (x, y) Î R 2 | y 2 + x 2 = 1, x> 0 }
15) f=( (x, y) Î R 2 | y2 + x2£ 1,x> 0 }
16) f=( (x, y) Î R 2 | x = y2 ,x³ 0 }
17) f=( (x, y) Î R 2 | y = sin(3x + p) )
18) f=( (x, y) Î R 2 | y = 1/cos x )
19) f=( (x, y) Î R 2 | y=2| x | + 3 )
20) f=( (x, y) Î R 2 | y=| 2x+1| )
21) f=( (x, y) Î R 2 | y = 3 x )
22) f=( (x, y) Î R 2 | y=e-x)
23) f =( (x, y)Î R 2 | y=e | x | )
24) f=( (x, y) Î R 2 | y = cos(3x) - 2 )
25) f=( (x, y) Î R 2 | y = 3x 2 - 2 )
26) f=( (x, y) Î R 2 | y = 1 / (x + 2) )
27) f=( (x, y) Î R 2 | y = ln(2x) - 2 )
28) f=( (x, y) Î R 2 | y=| 4x -1| + 2 )
29) f=( (x, y) Î R 2 | y = 1 / (x 2 + 2x-5))
30) f=( (x, y) Î R 2 | x = y 3, y³ - 2 }.
test questions
2. Definition of a binary relation.
3. Methods for describing binary relations.
4.Scope of definition and range of values.
5. Properties of binary relations.
6. Equivalence relation and equivalence classes.
7. Order relations: strict and non-strict, full and partial.
8. Classes of residues modulo m.
9.Functional relations.
10. Injection, surjection, bijection.
Lab #3
Relations. Basic concepts and definitions
Definition 2.1.ordered pair<x, y> is the collection of two elements x and y arranged in a certain order.
Two ordered pairs<x, y> and<u, v> are equal to each other if and only if x = u and y= v.
Example 2.1.
<a, b>, <1, 2>, <x, 4> are ordered pairs.
Similarly, we can consider triples, quadruples, n-ki elements<x 1 , x 2 ,…xn>.
Definition 2.2.Direct(or Cartesian)work two sets A and B is a set of ordered pairs such that the first element of each pair belongs to the set A, and the second - to the set B:
A ´ B = {<a, b>, ç aÎ BUT and bÏ AT}.
In general, the direct product n sets BUT 1 ,BUT 2 ,…A n is called a set BUT one BUT 2 ´ …´ A n, consisting of ordered sets of elements<a 1 , a 2 , …,a n> length n, such that i- th a i belongs to the set A i,a i Î A i.
Example 2.2.
Let be BUT = {1, 2}, AT = {2, 3}.
Then A ´ B = {<1, 2>, <1, 3>,<2, 2>,<2, 3>}.
Example 2.3.
Let be BUT= {x ç0 £ x£ 1) and B= {yç2 £ y£3)
Then A ´ B = {<x, y >, ç0 £ x£1&2 £ y£3).
Thus, the set A ´ B consists of points lying inside and on the border of a rectangle formed by straight lines x= 0 (y-axis), x= 1,y= 2and y = 3.
The French mathematician and philosopher Descartes was the first to propose a coordinate representation of points in a plane. This is historically the first example of a direct work.
Definition 2.3.binary(or double)ratio r is called the set of ordered pairs.
If a couple<x, y> belongs r, then it is written as follows:<x, y> Î r or, which is the same, xr y.
Example2.4.
r = {<1, 1>, <1, 2>, <2, 3>}
Similarly, one can define n-local relation as a set of ordered n-OK.
Since a binary relation is a set, the ways of specifying a binary relation are the same as the ways of specifying a set (see Section 1.1). A binary relation can be specified by enumerating ordered pairs or by specifying a general property of ordered pairs.
Example 2.5.
1. r = {<1, 2>, <2, 1>, <2, 3>) – relation is given by enumeration of ordered pairs;
2. r = {<x, y> ç x+ y = 7, x, y are real numbers) – the ratio is specified by specifying the property x+ y = 7.
In addition, a binary relation can be given binary relation matrix. Let be BUT = {a 1 , a 2 , …, a n) is a finite set. binary relation matrix C is a square matrix of order n, whose elements c ij are defined as follows:
Example 2.6.
BUT= (1, 2, 3, 4). Let's define a binary relation r in the three ways listed.
1. r = {<1, 2>, <1, 3>, <1, 4>, <2, 3>, <2, 4>, <3, 4>) – the relation is given by enumeration of all ordered pairs.
2. r = {<a i, a j> ç a i < a j; a i, a jÎ BUT) – the relation is specified by specifying the "less than" property on the set BUT.
3. - the relation is given by the matrix of the binary relation C.
Example 2.7.
Let's consider some binary relations.
1. Relations on the set of natural numbers.
a) relation £ holds for pairs<1, 2>, <5, 5>, but is not satisfied for the pair<4, 3>;
b) the relation "have a common divisor other than one" holds for pairs<3, 6>, <7, 42>, <21, 15>, but is not satisfied for the pair<3, 28>.
2. Relations on the set of points of the real plane.
a) the relation "to be at the same distance from the point (0, 0)" holds for the points (3, 4) and (–2, Ö21), but does not hold for the points (1, 2) and (5, 3);
b) the relation "to be symmetrical about the axis OY" is performed for all points ( x, y) and (- x, –y).
3. Relationships on a variety of people.
a) the attitude "to live in one city";
b) the attitude "to study in one group";
c) the attitude "to be older".
Definition 2.4. The domain of a binary relation r is the set D r = (x ç there is y such that xr y).
Definition 2.5. The range of a binary relation r is the set R r = (y çthere is x such that xr y).
Definition 2.6. The domain of a binary relation r is the set M r = D r ÈR r .
Using the concept of a direct product, we can write:
rÎ D r´ R r
If a D r= R r = A, then we say that the binary relation r set on the set A.
Example 2.8.
Let be r = {<1, 3>, <3, 3>, <4, 2>}.
Then D r ={1, 3, 4}, R r = {3, 2}, M r= {1, 2, 3, 4}.
Operations on relations
Since relations are sets, all operations on sets are valid for relations.
Example 2.9.
r 1 = {<1, 2>, <2, 3>, <3, 4>}.
r 2 = {<1, 2>, <1, 3>, <2, 4>}.
r 1 È r 2 = {<1, 2>, <1, 3>, <2, 3>, <2, 4>, <3, 4>}.
r 1 Z r 2 = {<1, 2>}.
r 1 \ r 2 = {<2, 3>, <3, 4>}.
Example 2.10.
Let be R is the set of real numbers. Consider the following relations on this set:
r 1 - "£"; r 2 – " = "; r 3 – " < "; r 4 - "³"; r 5 – " > ".
r 1 = r 2 È r 3 ;
r 2 = r 1 Z r 4 ;
r 3 = r 1 \ r 2 ;
r 1 = ;
We define two more operations on relations.
Definition 2.7. The relation is called reverse to the attitude r(denoted r- 1) if
r- 1 = {<x, y> ç< y, x> Î r}.
Example 2.11.
r = {<1, 2>, <2, 3>, <3, 4>}.
r- 1 = {<2, 1>, <3, 2>, <4, 3>}.
Example 2.12.
r = {<x, y> ç x – y = 2, x, y Î R}.
r- 1 = {<x, y> ç< y, x> Î r} = r- 1 = {<x, y> ç y–x = 2, x, y Î R} = {<x, y> ç– x+ y = 2, x, y Î R}.
Definition 2.8.Composition of two ratios r and s is called the ratio
s r= {<x, z> çthere is such y, what<x, y> Î r and< y, z> Î s}.
Example 2.13.
r = {<x, y> ç y = sinx}.
s= {<x, y> ç y = Ö x}.
s r= {<x, z> çthere is such y, what<x, y> Î r and< y, z> Î s} = {<x, z> çthere is such y, what y = sinx and z= Ö y} = {<x, z> ç z= Ö sinx}.
The definition of the composition of two relations corresponds to the definition of a complex function:
y = f(x), z= g(y) Þ z= g(f(x)).
Example 2.14.
r = {<1, 1>, <1, 2>, <1, 3>, <3, 1>}.
s = {<1, 2>, <1, 3>, <2, 2>, <3, 2>, <3, 3>}.
Finding process s r in accordance with the definition of composition, it is convenient to represent it as a table in which the enumeration of all possible values is implemented x, y, z. for every pair<x, y> Î r consider all possible pairs< y, z> Î s(Table 2.1).
Table 2.1
| <x, y> Î r | < y, z> Î s | <x, z> Î s r |
| <1, 1> <1, 1> <1, 2> <1, 3> <1, 3> <3, 1> <3, 1> | <1, 2> <1, 3> <2, 2> <3, 2> <3, 3> <1, 2> <1, 3> | <1, 2> <1, 3> <1, 2> <1, 2> <1, 3> <3, 2> <3, 3> |
Note that the first, third and fourth, as well as the second and fifth rows of the last column of the table contain identical pairs. So we get:
s r= {<1, 2>, <1, 3>, <3, 2>, <3, 3>}.
Relationship Properties
Definition 2.9. Attitude r called reflective on the set X, if for any xÎ X performed xr x.
It follows from the definition that any element<x,x > Î r.
Example 2.15.
a) Let X is a finite set X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <2, 2>, <3, 1>, <3, 3>). Attitude r reflexively. If a X is a finite set, then the main diagonal of the reflexive relation matrix contains only ones. For our example
b) Let X r relation of equality. This relation is reflexive, since each number is equal to itself.
c) Let X- lots of people and r attitude "to live in one city". This relation is reflexive, since everyone lives in the same city with himself.
Definition 2.10. Attitude r called symmetrical on the set X, if for any x, yÎ X from xry should yr x.
It's obvious that r symmetrical if and only if r = r- 1 .
Example 2.16.
a) Let X is a finite set X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <1, 3>, <2, 1>, <3, 1>, <3, 3>). Attitude r symmetrical. If a X is a finite set, then the symmetric ratio matrix is symmetric with respect to the main diagonal. For our example
b) Let X is the set of real numbers and r relation of equality. This relation is symmetrical, since if x equals y, then and y equals x.
c) Let X- many students and r the attitude of "learning in one group". This relation is symmetrical, since if x studying in the same group y, then and y studying in the same group x.
Definition 2.11. Attitude r called transitive on the set X, if for any x, y,zÎ X from xry and yrz should xrz.
Simultaneous fulfillment of conditions xry, yrz, xrz means that a couple<x,z>belongs to composition r r. Therefore, for transitivity r necessary and sufficient that the set r r was a subset r, i.e. r rÍ r.
Example 2.17.
a) Let X is a finite set X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <2, 3>, <1, 3>). Attitude r is transitive, because along with the pairs<x,y>and<y,z>has a couple<x,z>. For example, along with the pairs<1, 2>, and<2, 3>there is a couple<1, 3>.
b) Let X is the set of real numbers and r relation £ (less than or equal). This relation is transitive, since if x£ y and y£ z, then x£ z.
c) Let X- lots of people and r the attitude of being older. This relation is transitive, since if x older y and y older z, then x older z.
Definition 2.12. Attitude r called equivalence relation on the set X, if it is reflexive, symmetric and transitive on the set X.
Example 2.18.
a) Let X is a finite set X= (1, 2, 3) and r = {<1, 1>, <2, 2>, <3, 3>). Attitude r is an equivalence relation.
b) Let X is the set of real numbers and r relation of equality. This is an equivalence relation.
c) Let X- many students and r the attitude of "learning in one group". This is an equivalence relation.
Let be r X.
Definition 2.13. Let be r is the equivalence relation on the set X and xÎ X. Equivalence class, generated by the element x, is called a subset of the set X, consisting of those elements yÎ X, for which xry. Equivalence class generated by element x, denoted by [ x].
Thus, [ x] = {yÎ X|xry}.
The equivalence classes form partition sets X, i.e., a system of non-empty pairwise disjoint subsets whose union coincides with the entire set X.
Example 2.19.
a) The relation of equality on the set of integers generates the following equivalence classes: for any element x from this set [ x] = {x), i.e. each equivalence class consists of one element.
b) The equivalence class generated by the pair<x, y> is determined by the ratio:
[<x, y>] = .
Each equivalence class generated by a pair<x, y> defines one rational number.
c) For the relation of belonging to one student group, the equivalence class is the set of students of one group.
Definition 2.14. Attitude r called antisymmetric on the set X, if for any x, yÎ X from xry and yr x should x = y.
It follows from the definition of antisymmetry that whenever a pair<x,y> owned at the same time r and r- 1 , the equality x = y. In other words, r Ç r- 1 consists only of pairs of the form<x,x >.
Example 2.20.
a) Let X is a finite set X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <1, 3>, <2, 2>, <2, 3>, <3, 3>). Attitude r antisymmetric.
Attitude s= {<1, 1>, <1, 2>, <1, 3>, <2, 1>, <2, 3>, <3, 3>) is not antisymmetric. For example,<1, 2> Î s, and<2, 1> Î s, but 1 ¹2.
b) Let X is the set of real numbers and r relation £ (less than or equal). This relation is antisymmetric, since if x £ y, and y £ x, then x = y.
Definition 2.15. Attitude r called partial order relation(or just a partial order) on the set X, if it is reflexive, antisymmetric and transitive on the set X. A bunch of X in this case it is called partially ordered, and this relation is often denoted by the symbol £, if this does not lead to misunderstanding.
The relation inverse to the relation of partial order will obviously be the relation of partial order.
Example 2.21.
a) Let X is a finite set X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <1, 3>, <2, 2>, <2, 3>, <3, 3>). Attitude r
b) Attitude BUTÍ AT on the set of subsets of some set U is a partial order relation.
c) The divisibility relation on the set of natural numbers is a partial order relation.
Functions. Basic concepts and definitions
In mathematical analysis, the following definition of a function is accepted.
Variable y is called a function of a variable x, if, according to some rule or law, each value x corresponds to one specific value y = f(x). Variable area x is called the scope of the function, and the scope of the variable y– range of function values. If one value x matches several (and even infinitely many) values y), then the function is called multivalued. However, in the course of analysis of functions of real variables one avoids many-valued functions and considers single-valued functions.
Consider another definition of a function in terms of relationships.
Definition 2.16. Function is any binary relation that does not contain two pairs with equal first components and different second ones.
This relationship property is called uniqueness or functionality.
Example 2.22.
a) (<1, 2>, <3, 4>, <4, 4>, <5, 6>) is a function.
b) (<x, y>: x, y Î R, y = x 2 ) is a function.
in) (<1, 2>, <1, 4>, <4, 4>, <5, 6>) is a relation, not a function.
Definition 2.17. If a f is a function, then D f – domain, a Rf – range functions f.
Example 2.23.
For example 2.22 a) D f – {1, 3, 4, 5}; Rf – {2, 4, 6}.
For example 2.22 b) D f = Rf = (–¥, ¥).
Each element x D f function matches the only one element y Rf. This is denoted by the well-known notation y = f(x). Element x called a function argument or element preimage y with the function f, and the element y function value f on the x or element image x at f.
So, from all relations, functions are distinguished by the fact that each element from the domain of definition has the only one image.
Definition 2.18. If a D f = X and Rf = Y, then we say that the function f determined on X and takes its values on Y, a f called mapping of the set X onto Y(X ® Y).
Definition 2.19. Functions f and g are equal if their domain of definition is the same set D, and for any x Î D fair equality f(x) = g(x).
This definition does not contradict the definition of equality of functions as equality of sets (after all, we have defined a function as a relation, i.e., a set): sets f and g are equal if and only if they consist of the same elements.
Definition 2.20. Function (display) f called surjective or simply surjection, if for any element y Y element exists x Î X, such that y = f(x).
So every function f is a surjective mapping (surjection) D f® Rf.
If a f is a surjection, and X and Y are finite sets, then ³ .
Definition 2.21. Function (display) f called injective or simply injection or one-to-one if from f(a) = f(b) should a = b.
Definition 2.22. Function (display) f called bijective or simply bijection if it is both injective and surjective.
If a f is a bijection, and X and Y are finite sets, then = .
Definition 2.23. If the range of the function D f consists of one element f called constant function.
Example 2.24.
a) f(x) = x 2 is a mapping of the set of real numbers onto the set of non-negative real numbers. Because f(–a) = f(a), and a ¹ – a, then this function is not an injection.
b) For each x R= (– , ) function f(x) = 5 is a constant function. It displays many R to the set (5). This function is surjective, but not injective.
in) f(x) = 2x+ 1 is an injection and a bijection, because from 2 x 1 +1 = 2x 2+1 follows x 1 = x 2 .
Definition 2.24. The function that implements the display X one X 2 ´...´ X n ® Y called n-local function.
Example 2.25.
a) Addition, subtraction, multiplication and division are binary functions on the set R real numbers, that is, functions of the type R 2® R.
b) f(x, y) = is a two-place function that implements the mapping R ´ ( R \ )® R. This function is not an injection, because f(1, 2) = f(2, 4).
c) The lottery payoff table defines a two-place function that establishes a correspondence between pairs from N 2 (N is the set of natural numbers) and the set of payoffs.
Since functions are binary relations, it is possible to find inverse functions and apply the composition operation. Composition of any two functions is a function, but not for every function f attitude f-1 is a function.
Example 2.26.
a) f = {<1, 2>, <2, 3>, <3, 4>, <4, 2>) is a function.
Attitude f –1 = {<2, 1>, <3, 2>, <4, 3>, <2, 4>) is not a function.
b) g = {<1, a>, <2, b>, <3, c>, <4, D>) is a function.
g -1 = {<a, 1>, <b, 2>, <c, 3>, <D, 4>) is also a function.
c) Find the composition of functions f from example a) and g-1 from example b). We have g -1f = {<a, 2>, <b, 3>, <c, 4>, <d, 2>}.
fg-1 = Æ.
Notice, that ( g -1f)(a) = f(g -1 (a)) = f(1) = 2; (g -1f)(c) = f(g -1 (c)) = f(3) = 4.
An elementary function in mathematical analysis is any function f, which is a composition of a finite number of arithmetic functions, as well as the following functions:
1) Fractional-rational functions, i.e. functions of the form
a 0 + a 1 x + ... + a n x n
b 0 + b 1 x + ... + b m x m.
2) Power function f(x) = x m, where m is any constant real number.
3) exponential function f(x) = ex.
4) logarithmic function f(x) = log x, a >0, a 1.
5) Trigonometric functions sin, cos, tg, ctg, sec, csc.
6) Hyperbolic functions sh, ch, th, cth.
7) Inverse trigonometric functions arc sin, arccos etc.
For example, the function log 2 (x 3 +sincos 3x) is elementary, because it is the composition of functions cosx, sinx, x 3 , x 1 + x 2 , logx, x 2 .
An expression describing the composition of functions is called a formula.
For a multiplace function, the following important result is valid, which was obtained by A. N. Kolmogorov and V. I. Arnold in 1957 and is a solution to the 13th Hilbert problem:
Theorem. Every continuous function n variables can be represented as a composition of continuous functions of two variables.
Ways to set functions
1. The easiest way to set functions is tables (Table 2.2):
Table 2.2
However, functions defined on finite sets can be defined in this way.
If a function defined on an infinite set (segment, interval) is specified at a finite number of points, for example, in the form of trigonometric tables, tables of special functions, etc., then interpolation rules are used to calculate the values of functions at intermediate points.
2. A function can be defined as a formula that describes a function as a composition of other functions. The formula specifies the sequence in which the function is calculated.
Example 2.28.
f(x) = sin(x + Ö x) is a composition of the following functions:
g(y) = Ö y; h(u, v) = u+v; w(z) = sinz.
3. The function can be given in the form recursive procedure. The recursive procedure defines a function defined on the set of natural numbers, i.e. f(n), n= 1, 2,... as follows: a) the value f(1) (or f(0)); b) meaning f(n+ 1) is defined through composition f(n) and other well-known functions. The simplest example of a recursive procedure is the calculation n!: a) 0! = 1; b) ( n + 1)! = n!(n+ 1). Many numerical method procedures are recursive procedures.
4. There are ways to define a function that do not contain a way to calculate the function, but only describe it. For example:
f M(x) =
Function f M(x) is the characteristic function of the set M.
So, according to the meaning of our definition, define the function f- means to set the display X ® Y, i.e. define a set X´ Y, so the question is reduced to specifying some set. However, it is possible to define the concept of a function without using the language of set theory, namely: a function is considered given if a computational procedure is given that, given the value of the argument, finds the corresponding value of the function. A function defined in this way is called computable.
Example 2.29.
Determination procedure Fibonacci numbers, is given by the relation
F n= Fn- 1 + Fn- 2 (n³ 2) (2.1)
with initial values F 0 = 1, F 1 = 1.
Formula (2.1), together with the initial values, determines the following series of Fibonacci numbers:
| n | 0 1 2 3 4 5 6 7 8 9 10 11 … |
| F n | 1 1 2 3 5 8 13 21 34 55 89 144 … |
The computational procedure for determining the value of a function from a given argument value is nothing more than algorithm.
Security questions for topic 2
1. Specify the ways of specifying a binary relation.
2. The main diagonal of the matrix of which ratio contains only ones?
3. For what relationship r condition is always met r = r- 1 ?
4. For what relationship r condition is always met r rÍ r.
5. Introduce equivalence and partial order relations on the set of all lines in the plane.
6. Specify ways to set functions.
7. Which of the following statements is true?
a) Every binary relation is a function.
b) Every function is a binary relation.
Topic 3. GRAPHS
Euler's first work on graph theory appeared in 1736. In the beginning, this theory was associated with mathematical puzzles and games. However, later graph theory began to be used in topology, algebra, and number theory. Nowadays, graph theory finds application in a wide variety of fields of science, technology and practice. It is used in the design of electrical networks, planning transportation, building molecular schemes. Graph theory is also used in economics, psychology, sociology, and biology.
In this subsection, we introduce Cartesian products, relations, functions, and graphs. We study the properties of these mathematical models and the connections between them.
Cartesian product and enumeration of its elements
Cartesian product sets A and B is called a set consisting of ordered pairs: A´ B= {(a,b): (aÎ A) & (bÎ B)}.
For sets A 1, …, A n the Cartesian product is defined by induction:
In the case of an arbitrary set of indices I cartesian product families sets ( Ai} i Î I is defined as a set consisting of such functions f:I® A i , what is for everyone iÎ I right f(i)Î Ai .
Theorem 1
Let be A andB are finite sets. Then |A´ B| = |A|×| B|.
Proof
Let be A = (a 1 , …,am), B=(b 1 , …,bn). The elements of a Cartesian product can be arranged using a table
(a 1 ,b 1), (a 1 ,b 2), …, (a 1 ,b n);
(a 2 ,b 1), (a 2 ,b 2), …, (a 2 ,b n);
(a m ,b 1), (a m ,b 2),…, (a m ,b n),
consisting of n columns, each of which consists of m elements. From here | A´ B|=mn.
Corollary 1
Proof
With the help of induction on n. Let the formula be true for n. Then
Relations
Let be n³1 is a positive integer and A 1, …, A n are arbitrary sets. The relationship between the elements of the sets A 1, …, A n or n-ary relation is called an arbitrary subset.
Binary relations and functions
binary relation between elements of sets A and B(or, for short, between A and B) is called a subset RÍ A´ B.
Definition 1
Function or mapping is called a triple consisting of sets A and B and subsets fÍ A´ B(function graph) satisfying the following two conditions;
1) for any xÎ A there is such yÎ f, what (x,y)Î f;
2) if (x,y)Î f and (x,z)Î f, then y=z.
It is easy to see that fÍ A´ B will define a function if and only if for any xÎ A there is only one yÎ f, what ( x,y) Î f. This y denote by f(x).
The function is called injection, if for any x,x'Î A, such what x¹ x', takes place f(x)¹ f(x'). The function is called surjection if for each yÎ B there is such xÎ A, what f(x) = y. If a function is an injection and a surjection, then it is called bijection.
Theorem 2
For a function to be a bijection, it is necessary and sufficient that there exists a function such that fg =ID B and gf =ID A.
Proof
Let be f- bijection. Because of the surjectivity f for everybody yÎ B you can select an element xÎ A, for which f(x) = y. Due to the injectivity f, this element will be the only one, and we will denote it by g(y) = x. Let's get a function.
By function construction g, there are equalities f(g(y)) = y and g(f(x)) = x. So that's right fg =ID B and gf =ID A. The converse is obvious: if fg =ID B and gf =ID A, then f– surjection into force f(g(y)) = y, for everybody yÎ B. In this case, from will follow 
, which means . Hence, f- injection. Hence it follows that f- bijection.
Image and prototype
Let be a function. way subsets XÍ A called a subset f(X) = (f(x):xÎ x)Í b. For YÍ B subset f - -1 (Y) =(xÎ A:f(x)Î Y) called prototype subsetsY.
Relationships and Graphs
Binary relationships can be visualized using directed graphs.
Definition 2
directed graph is called a pair of sets (E,v) along with a couple of displays s,t:E® V. Set elements V are represented by points on a plane and are called peaks. Items from E are called directed edges or arrows. Each element eÎ E depicted as an arrow (possibly curvilinear) connecting the vertex s(e) top t(e).
Arbitrary binary relation RÍ V´ V corresponds to a directed graph with vertices vÎ V, whose arrows are ordered pairs (u,v)Î R. Displays s,t:R® V are determined by the formulas:
s(u,v) =u and t(u,v) =v.
Example 1
Let be V = (1,2,3,4).
Consider the relation
R = ((1.1), (1.3), (1.4), (2.2), (2.3), (2.4), (3.3), (4.4)).
It will correspond to a directed graph (Fig. 1.2). The arrows of this graph will be pairs (i,j)Î R.
Rice. 1.2. Directed binary relation graph
In the resulting directed graph, any pair of vertices is connected by at most one arrow. Such directed graphs are called simple. If we do not consider the direction of the arrows, then we arrive at the following definition:
Definition 3
Simple (undirected) graph G = (V,e) is called a pair consisting of a set V and many E, consisting of some unordered pairs ( v 1 ,v2) elements v 1 ,v2Î V such that v1¹ v2. These couples are called ribs, and elements from V – peaks.
Rice. 1.3. Simple undirected graph K 4
A bunch of E defines a binary symmetric antireflexive relation consisting of pairs ( v 1 ,v2), for which ( v 1 ,v2} Î E. The vertices of a simple graph are shown as points, and the edges as line segments. On fig. 1.3 shows a simple graph with many vertices
V ={1, 2, 3, 4}
and many ribs
E= {{1,2}, {1,3},{1,4}, {2,3}, {2,4}, {3, 4}}.
Operations on binary relations
binary relation between elements of sets A and B called an arbitrary subset RÍ A´ B. Recording aRb(at aÎ A, bÎ B) means that (a,b)Î R.
The following relational operations are defined RÍ A´ A:
· R-1= ((a,b): (b,a)Î R);
· R° S = ((a, b): ($ xÎ A)(a, x)Î R & (x,b)Î R);
· R n =R°(Rn-1);
Let be Id A = ((a,a):aÎ A)- identical relationship. Attitude R Í X´ X called:
1) reflective, if (a,a)Î R for all aÎ X;
2) anti-reflexive, if (a,a)Ï R for all aÎ X;
3) symmetrical if for all a,bÎ X the implication is correct aRbÞ bRa;
4) antisymmetric, if aRb &bRaÞ a=b;
5) transitive if for all a,b,cÎ X the implication is correct aRb &bRcÞ aRc;
6) linear, for all a,bÎ X the implication is correct a¹ bÞ aRbÚ bRa.
Denote ID A through ID. It is easy to see that the following holds.
Suggestion 1
Attitude RÍ X´ X:
1) reflexively Û IDÍ R;
2) antireflexively Û RÇ Id=Æ ;
3) symmetrically Û R=R-1;
4) antisymmetric Û RÇ R-1Í ID;
5) transitively Û R° RÍ R;
6) linearly Û RÈ IDÈ R-1 = X´ X.
binary relation matrix
Let be A= {a 1, a 2, …, a m) and B= {b 1, b 2, …, b n) are finite sets. binary relation matrix R Í A ´ B is called a matrix with coefficients:
Let be A is a finite set, | A| = n and B= A. Consider the algorithm for calculating the composition matrix T= R° S relations R, S Í A´ A. Denote the coefficients of the relationship matrices R, S and T respectively through rij, sij and tij.
Since the property ( a i,a k)Î T is tantamount to the existence of such a jÎ A, what ( a i,a j)Î R and ( a j,a k) Î S, then the coefficient tik will be equal to 1 if and only if such an index exists j, what rij= 1 and sjk= 1. In other cases tik equals 0. Therefore, tik= 1 if and only if .
This implies that to find the composition matrix of relations, it is necessary to multiply these matrices and, in the resulting product of matrices, replace non-zero coefficients with units. The following example shows how the composition matrix is calculated in this way.
Example 2
Consider a binary relation on A = (1,2,3) equal to R = ((1,2),(2,3)). Let's write the relation matrix R. By definition, it consists of the coefficients r 12 = 1, r23 = 1 and others rij= 0. Hence the relation matrix R is equal to:
Let's find the relation R° R. To this end, we multiply the ratio matrix R to myself:
.
We get the relation matrix:
Hence, R° R= {(1,2),(1,3),(2,3)}.
Proposition 1 implies the following corollary.
Consequence 2
If a A= B, then the ratio R on the A:
1) reflexively if and only if all elements of the main diagonal of the relation matrix R are equal to 1;
2) antireflexively if and only if all elements of the main diagonal of the relation matrix R are 0;
3) symmetrical if and only if the relation matrix R symmetrical;
4) transitively if and only if each coefficient of the relation matrix R° R not greater than the corresponding coefficient of the ratio matrix R.
Communication has always been seen as polyfunctional process. Psychologists define the functions of communication according to different criteria: emotional, informational, socializing, connecting, translational, aimed at self-knowledge (A. V. Mudrik), establishing community, self-determination (A. B. Dobrovich), self-expression (A. A. Brudny), cohesion, etc. Most often in psychology, the functions of communication are considered in accordance with the model of relations "man-activity-society".
Five main functions can be distinguished: pragmatic, formative, confirming, organization and maintenance of interpersonal relationships, intrapersonal (Fig. 7).
AT pragmatic function Communication acts as the most important condition for bringing people together in the process of any joint activity. The famous biblical story about the construction of the Tower of Babel tells about the destructive consequences for the activities of people if this condition is not met.
Rice. 7.
A big role belongs formative function communication. Communication between a child and an adult is not just a process of transferring to the first the sum of skills, skills and knowledge that he mechanically learns, but a complex process of mutual influence, enrichment and change. The vital role of communication is clearly shown in the following example. In the 30s. 20th century in the United States, an experiment was conducted in two clinics in which children were treated for serious, poorly curable diseases. The conditions in both clinics were the same, but with some differences: in one hospital, relatives were not allowed to see the babies, for fear of infection, and in the other, at certain hours, parents could talk and play with the child in a specially designated room. After a few months, the effectiveness of treatment was compared. In the first department, the mortality rate approached one-third, despite the efforts of doctors. In the second department, where babies were treated with the same means and methods, not a single child died.
Confirmation function in the process of communication makes it possible to know, to affirm oneself. Wanting to establish himself in his existence and his value, a person is looking for a foothold in another person. Everyday experience of human communication is replete with procedures organized according to the principle of confirmation: rituals of acquaintance, greetings, naming, giving various signs of attention. The famous English psychiatrist R. D. Laing saw in non-confirmation a universal source of many mental illnesses, primarily schizophrenia.
interpersonal for any person is associated with the evaluation of people and the establishment of certain emotional relationships - either positive or negative. Therefore, an emotional attitude towards another person can be expressed in terms of "sympathy - antipathy", which leaves its mark not only on personal, but also on business communication.
intrapersonal function considered as a universal way of human thinking. L. S. Vygotsky noted in this connection that "a person, even alone with himself, retains the function of communication."
So, the leading significance of communication in human life is that it is a means of organizing the joint activities of people and a way to satisfy a person's need for another person, their live contact.
Communication as a socio-psychological phenomenon is the contact between people, which is carried out through language and speech, has different forms of manifestation. Language is a system of verbal signs, a means by which communication is carried out between people. The use of language to communicate between people is called speech. Depending on the characteristics of communication, its various types are distinguished (Fig. 8).
By contact with the interlocutor, communication can be direct and indirect.
Direct communication (direct) - this is natural communication, when the subjects of interaction are nearby and communicate through speech, facial expressions and gestures.
Rice. eight.
This type of communication is the most complete, because in the process of it, individuals receive maximum information about each other.
Mediated (indirect) communication carried out in situations where individuals are separated from each other by time or distance. For example: talking on the phone, correspondence. Mediated communication is incomplete psychological contact when feedback is difficult.
Communication can be interpersonal or mass. Mass communication represents multiple contacts of strangers, as well as communication mediated by various types of mass media. It may be direct and mediated. direct mass communication observed at rallies, meetings, demonstrations, in all large social groups: the crowd, the public, the audience. Mediated mass communication has a one-sided character and is associated with mass culture and mass media.
According to the criterion of equality of partners in interpersonal communication (Fig. 9), two types are distinguished: dialogic and monologue.
Dialogical communication- equal subject-subject interaction, aimed at mutual knowledge, the desire to achieve the goals of each partner.
monologue communication is realized with unequal positions of partners and represents a subject-object relationship. It can be imperative and manipulative. imperative communication- an authoritarian, directive form of interaction with a partner in order to achieve control over his behavior, attitudes, thoughts and coercion to certain actions or decisions. Moreover, this goal is not veiled. manipulative communication- a form of interpersonal communication in which the impact on the communication partner is carried out covertly to achieve their intentions.
Rice. nine.
There are two types of communication - role and personal. AT role communication people act based on their status. For example, communication between a teacher and students, a shop manager with workers, etc. will be role-playing. Role communication is regulated by the rules accepted in society and the specifics of treatment. personal communication depends on the individual characteristics of people and the relationship between them.
Communication can be short-term or long-term, depending on the goals, the content of the activity, the individual characteristics of the interlocutors, their likes, dislikes, etc.
The exchange of information can occur through verbal and non-verbal interaction. Verbal communication happens through speech non-verbal- with the help of paralinguistic means of transmitting information (loudness of speech, timbre of voice, gestures, facial expressions, postures).
Communication takes place at different levels. The levels of communication are determined by the general culture of the interacting objects, their individual and personal characteristics, the peculiarities of the situation, social control, the value orientations of those who communicate, their relationship to each other (Fig. 10).
Rice. ten.
The most primitive level of communication phatic(from lat. fatuus - stupid). It involves a simple exchange of remarks to maintain a conversation, does not have a deep meaning. Such communication is necessary in standardized conditions or is determined by etiquette norms.
Informational the level of communication involves the exchange of interesting new information for the interlocutors, which is a source of emotional, mental, behavioral activity of a person.
personal the level of communication characterizes such an interaction in which subjects are capable of deep self-disclosure and comprehension of the essence of another person, themselves and the world around them. It is built on a positive attitude towards yourself, other people and the world around you as a whole. This is the highest spiritual level of communication.
Loading...