Topic : Algebra of sets.

We study different types of algebra: school algebra of rational numbers, linear algebra, vector algebra, etc.

Question: " What is an algebra? "

Answer: An algebra is a set of elements of an arbitrary nature together with a number of operations defined on the elements of the given set. The nature of the set elements, the number and the properties of the operations determine the specific type of the algebra.

In the previous topic we defined some operations on sets. These operations have certain properties, quite similar to those of arithmetic operations.

Properties of arithmetic operations:

Properties x + 0 = x (x plus zero is x)
of zero: x * 0 = 0 (x multiplied by zero is zero)

Commutative x + y = y + x (x plus y is equal to y + x)
laws: x * y = y * x [ Kə ' m j u: t a t i v ]

Associative (x + y) + z = x +(y+z)

laws: (x * y) * z = x * (y * z) [ə 'souʃ iə tiv]

Distribitive law: x * (y + z) = x * y + x *

Union and intersection of sets are subject to such laws:

1. Laws of simplification:
a)A U A = A; b)A ∩ A = A;

2. Commutative laws:

a)A U B = B U A; b)A ∩ B = B ∩ A;

3. Associative laws:

a)A U (B U C) = (A U B)U C;

b)A ∩ (B ∩ C) = (A ∩ B) ∩ C;

4. Distributive laws:

a)A ∩ (B U C) = (A ∩ B)U(A ∩ C); b)A U(B ∩ C) = (A U B) ∩ (A U C);

5. Elimination laws:

a) A U(A ∩ B) = A; b) A ∩ (A U B) = A

6) Laws for constants:

a)A U Ù = Ù; A ∩ Ø = Ø;

b) A U Ø = A; A ∩ Ù = A;

It is possible tо prove these laws using Wenn diagrams.

For example, the 1st distributive law:

А ∩ (B U C) (A ∩ B)U(A ∩ C)

The set compliment operation is subject to such laws:

7) The law of double compliment:

A = A;

8) The law of the excluded middle:

A U Ā = Ù;

9) The law of contradiction:

A ∩ Ā = Ø;

10) De Morgan's laws:

а) = Ā ∩ b) = Ā U ;

All these laws variables A, B, C,..... represent not just individual sets but arbitrary formulas of the algebra of sets.

All these laws can be used for equivalent transformations of the formulas of the set algebra usually in order to simplify them.

For example, simplify the formula:

A U B ∩ = A U B ∩ (Ā U ) = A U (B ∩ Ā U B ∩ ) = A U(B ∩ Ā)= = (A U B)∩ (A U Ā)=A U B;

H/t 5 Prove correctness of the second distributive law and the first De Mrgan's law, using Wenn diagrams.

H/t 6 Simplify the formula:

H/t 7. Simpify the formula: (Ā - B)U(A ∩ )

In formulas some brackets may be omitted if we use the rules of precedence of operations: -, ∩, U.

Ā U B ∩ C = (Ā)U(B ∩ C)

H/t 6.Solution: (Ā - B)U(A ∩ ) = (A∩ ) U (A ∩ ) =

(C U A)∩ (C U )(by the distributive law)=

(A ∩ U A)∩ (A ∩ U ) = A ∩

Topic: Relations.

The concept of a relation is one of the fundamental concepts of mathematics. This notion is used to indicate a link between objects or ideas. Examples of relations: to sit at the same table, to play in the same team, etc.

(A formal definition of the notion of a relation is based on the operation of Cartesian product of sets A and B).

Cartesian product of sets A and B (denoted by A ´ B) is a set of all ordered pairs (a, b) where a ∈ A, b ∈ B.

Example: A = { 1, 2}; B = {4, 5};

A ´ B = {(1, 4), (1, 5), (2, 4), (2, 5)}.

B ´ A = {(4, 1), (4, 2), (5, 1), (5, 2)}.

A ´ A = A² = {(1, 1), (1, 2), (2, 1), (2, 2)} (a squared set).

If A and B are finite sets, the cardinality |A ´ B| = |A| ´ |B|.

In similar way a Cartesian product of K sets is defined so:

A ´ B ´... ´ Z = {(a, b,..., z)| a ∈ A; b ∈ B,..., z ∈ Z}

as a set of all possible ordered sequences of the above specified form.

Cartesian power of a set: A ´ A ´..... ´ A = A^k.

K times

Any subset λ of A ´ B of arbitrary sets A and B is called a relation, defined on A and B.

Example: A = B ={1, 2, 3, 4}

A ´ A = A² = { (1, 1), (1, 2), (1, 3), (1, 4),

(2, 1), (2, 2), (2, 3), (2, 4),

(3, 1), (3, 2), (3, 3), (3, 4),

(4, 1), (4, 2), (4, 3), (4, 4)}

λ = {(2, 1), (3, 2), (4, 3)} - this is a relation a λ b which can be defined by mean of a natural language like this: " a is greater then b by 1".

If (a, b) ∈ λ, then the element a is said to be in relation λ to the element b.(The relation λ is true for a and b).

Sometimes instead (a, b) ∈ λ another notation is used: a λ b.

A relation defined on a pair of sets A and B is called a binary relation.

A relation defined on a triple of sets is called a ternary relation or a three-place relation.

For example:

A = {a set of scientific advisors}

B = {a set of students}

C = {a set of topics in scientific research}.

Then β ⊆ A ´ B ´ C can be chosen so that it will represent the following relation:

“a is a scientific advisor of a student b in his research on a tоpic c”.

In a similar way k-place relations are introduced.