Table 1.1.1 Laws of the algebra of sets

1		Idempotent laws
2		Associative laws
3		Commutative laws
4		Distributive laws
5		Identity laws
6	= A	Involution law
7		Complement laws
8		De Morgan's laws

2. Relations

2.1. Product Sets

Definition. A set is called an ordered set if to each element there correspons a number n(n ϵ N) and elements are listed in the increasing manner.

Definition. Let two arbitrary sets A and B be given. The set of all ordered pairs (a,b) where a ϵ A and b ϵ B is called the product, or Cartesian product, of the sets A and B. A short designation of this product is A x R, which is read ‘A cross B ". By definition

A x B = {(a,b)| a ϵ A and b ϵB }.

One frequently writes A² instead of A x A.

Example 2.1.1. R denotes the set of real numbers and so R² = R x R is the set or ordered pairs of real numbers. We are familiar with the geometrical representation

of R² as points in the plane. Each point P represents an ordered pair (a,b) of real numbers and vice versa; the vertical line through P meets the x - axis at a and the

horizontal line through P meets the y - axis at b. R² is frequently called the Cartesian plane.

This idea of a product of sets can be extended to any finite number of sets. For any sets A₂,...,An the set of all ordered n - tuples (a₂,...,an) where a₂ ϵ A₂,...,an ϵ An is called the product of sets A₂,...,An and is denoted

by A1*A2*…*An .

2.2. Binary Relations

Definition. A binary relation between elements of the sets A and B is any subsetR of the set A x B that is R c A x B.

Suppose R is a relation from A to B. Then R is a set of ordered pairs where each first element comes from A end each second element comes from B. That is, for each aϵ A and b ϵB, exactly one of the following is true:

1. (a,b) ϵ R; we then say " a is R - related to b written aRb ;

2. (a, b) ϵ R; we then say " a is not R - related to b written

If R is a relation from a set A to itself, that is, if R is a subset of A² = A x A, then we say that ^ is a relation on A

Definition. The domain of a relation R is the set of all first elements of the ordered pairs which belong to and the range is the set of second elements.

Then we have the binary relation between the sets A and B.

Example 2.2.1. Given A = {1,2,3} and B = {.x, y, z}, and let R= {(1, y), (1, z), (3, y)}. Then R is a relation from A to B since R is a subset of A x B. With respect to this relation, 1Ry, 1Rz, 3Ry. The domain of R is {1,3} and the range is {y, z}.

Example 2.2.2. Let us denote in the table the elements belonging to the set R = {(a,1), (b,m), (A,0)} of the Cartesian product of the sets A and R by the points (R ^ (A X B)):