**Methods for specifying a set.**

1. Enumeration: A={1,2,3,4,5};B={1,2,..,99}

2. With the help of a characteristic property – a property that any element included in the set has and an element not included in the set does not have: А={а|P(a)}; B={k|k-even number}

3. Specifying the generating procedure.

A generating procedure is a process that, when launched, generates all the elements of a given set. K={k|k=5n -3n-2, where n N}.

An enumeration can only define finite sets.

**Euler-Venn diagram.**

For a visual representation of sets and their properties, diagrams are used, while sets are thought of as a set of points on a circle.

Universal set: – contains all other sets (V) as subsets, it is customary to depict a rectangle or a set.

.y

M H M AT

.x x BUT

y AT

**2) Operations on sets and their properties.**

1. Combination (sum) A and B – A B is the set consisting of those and only those elements belonging to at least one of the sets A or B.

ODA: A B={x|x A or x AT}

2. Intersection (product) A and B – A B consists of elements belonging to both A and B

3. The difference between sets A and B is the set A B consisting of those and only those elements that belong to A and do not belong to B.

AB={x| X A and x B }=A

A special case of difference: if A V, VA= ={x|x A} is the complement of the set A.

4. The symmetric difference (ring sum) A and B is called the set A B (A C) consisting of elements of the union of these sets, but not included in the intersection of these sets.

BUT B=(A B)( A B)=(AB) (BA)={x| (X A and x B) or (x B and x BUT)}

5. The Cartesian product (direct) of sets A and B is called sets AxB consisting of all ordered pairs (a, b) where a A and in AT.

AxB={(a, c)| a A, in AT }

**Properties of operations on sets.**

1.Commutativity

BUT B=B A;B A=A AT

2.Associativity

(BUT AT) C=A (AT C);(A AT) C=A (AT WITH)

3. Distributivity

BUT ( AT C)=(A AT) (BUT C); A (AT C)=(A AT) (BUT WITH)

4. Idenpotency rule

BUT A=A;A A=A; BUT =A; BUT = ;BUT V= ;A V=A

5.Law absorption

BUT (BUT B) u003d A; A (BUT B)=A

6. De’Morgan’s law

= ; =

7.Law of double complement (involutiveness)

=A

8.Law incorporation

BUT AT

9. The law of equality

A=B (( BUT AT) ( AT BUT)) ( BUT AT) ( )

10.Supplement Properties

BUT =V; BUT =

**3) Binary relations.**

**Def:** n-place relation (predicate P) on sets A BUT …BUT is any subset of the Dewart product A x A x…xA .

If: n=1, then P is called unary and P(x ) BUT

n=2, then Р is called binary and this set of ordered pairs Р(х X ) BUT x A ,

X BUT X BUT

**def:** x …X are called coordinates or components of the relation P

**def** : And the id relation ={(x,x)/x A} is called the identity relation

V =A =AxA={(x,y)/ x A, y And } is called a complete ratio or square.

Let P be a binary relation of sets A and A i.e. R BUT Ha

**Def:** the domain of the binary relation P is the set D=Dom P={х/ y: (xy) R}. The range of the binary relation Р is the set R=Уm P= {х/ x: (xy) R}. Binary relations R and S are called equal (R=S) if (x,y) R (x,y) S i.e. when the ratios R and S are equal as sets. If there is an entry that (x,y) P, then we say that the elements x, y are related by the relation P, or that x is in the relation P with y, or that the relation P holds for x and y. (x, y) R xPy

**Ways to specify binary relations.**

1. Listing. Applicable only for finite sets

2.Characteristic property

3.chart

4. Graph (if A=I then the diagram becomes a graph). P we put in correspondence the following geom.figure: points yavl.Dom P, Um P, and oriented edges (lines) i.e. (a, c) P we associate an oriented edge going from A to B (A C) with a fixed entry direction. We will call such a figure a directed graph of the relation P. Each binary relation P on a finite set can be associated with a directed graph and vice versa.

P u003d {(a, c), (c, c), (d, d), (e, a), (e, e), (c, d)}

in

with

a d

e

5. Graph (this method is applicable if the relationship is set on numerical sets)

Graph P is the set of all points of the Oxy plane with coordinates (x, y) R

Example:

P={(x,y)/x,y R,y=x }

6. Table (for finite sets)

7. Matrix (considered __finite__ set A)

A={(a …a )}, B={in ,in …in }; R AhB -b.o

||R|| matrix b.o P is called ||P||=(P ) of size nxm, n=|A|, m=|B|

1 if (and ,in ) R

R ={

0 if (and ,in ) R

**4) Operations on relations.**

**Def** : the inverse relation to P (inversion) is called P ={(y,x)/(x,y) R} thus **def** . unary operation of transition to reverse rel.

Composition (superposition) b.o R AxB and Q BxC is called the set P Q={(x,y)/x A,y c, z B : (x,z) P,(z,y) Q}

For any b.o., the following properties hold:

1.( P ) =P

2.( P Q) =Q R

3. .( P Q) R=P (Q R)

**Holy Island b.o on the set.**

Let the relation R be given on a non-empty set A R A

1. Reflexivity: a b.o is called reflexive on A if X A, (x, x) R

R reflexively every vertex of the graph has a loop

2. Anti-reflexivity: b.o R on A is called anti-reflexive if X A, (x, x) R

R is antireflexive when each vertex does not contain a loop

3. Symmetry: b.o R on A is called symmetric if for x,y A, (x, y) R (y,x) R. R symmetrical when together with each edge (x, y) the graph contains an edge (y, x)

4. Antisemmetricity: b.o R on A is called antisemmetric if x,y A(x,y) R and (y,x) R x=y.

R-antisymmetric: if two distinct vertices of a graph are connected by an edge, then only 1; in this case, there can be loops in the graph

1) the divisibility relation on the set R

a:b and c:a a=b

2) The inclusion relation on any subset of a universal subset is an antisemmeter. BUT B and B BUT A=B

5. Asymmetry: b.o is asymmetric on A if for each pair of elements x and y from the set A the simultaneous fulfillment of the relations (x, y) R and (y, x) R is not possible i.e. x,y What if (x, y) R, then (y, x) RR asymmetric if the graph contains an edge (x, y), then it does not contain an edge (y, x)

6. Transitivity: b.o is called transitive on A if x,y,z R if (x,y) R and (y, z) Rto (x,z) R

R transit. when the graph together with each pair of last edges (x, y), (y, z) sod. Rubro closing (x, z)

7. Connectedness: a b.o. is called connected on A if x,y A: x y or (x, y) R or (y, x) R. R connected when any 2 vertices of the graph are connected by one and only one edge.

1)”>” is connected on R; on R disconnected

**5) Matrix representation of binary relations. Properties of the matrix of binary relations.**

A binary relation can be specified using a matrix

Consider a finite set A and B

A={a _{1} ,a _{2} ,…,a _{n} }, B={b _{1} ,b _{2} ,…,b _{n} }

P⊆AxB, P is a binary relation

**Def:** The matrix of a binary relation P is a matrix ||P||=(p _{ij} ) of dimension nxm, where n=|A|, m=|B|

This matrix contains complete information about the relationship between the elements of sets A and B and allows you to present this information in a graphical form.

Note that any matrix consisting of 0 and 1 is a matrix of some binary relation

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