**
Operations with sets
**

*
Designation of sets and their
elements. Equality of sets.
*

*
Subset
(
inclusion
).
Sum (
union
) of sets.
*

*
Product (
intersection
) of sets. Difference (
complement
) of sets.
*

*
Symmetric difference of sets.
Properties of operations with sets.
*

Sets are designated by capital
letters, and their elements – by small letters. The
record
*
a
R
*
means, that an element
*
à
*
belongs to a set
*
R
*
,
i.e.
*
à
*
is an element of the set
*
R
*
. Otherwise, if
*
à
*
doesn't belong to the set
*
R
*
, we write
*
a
*
*
R
*
.

Two sets
*
À
*
and
*
B
*
are called
**
equal
**
(

*À*=

*Â*), if they consist of the same elements, i.e. each element of the set

*A*is an element of the set

*B*and vice versa, each element of the set

*Â*is an element of the set

*A*.

We
say,
that a set
*
À
*
is
included in a set
*
Â
*
(
Fig.1
)
or the
set
*
A
*
is
a
**
subset
**
of the set

*B*

*( in this case we write*

*À*

*Â*), if each element of the set

*A*is an element of the set

*B*. This dependence between sets is called an

**. The inclusions**

*inclusion**À*and

*À*

*À*take place for each set

*A*.

**
A
sum (
union
)
of sets
**

*À*and

*Â*( it's written as

*À*

*Â*) is a set of elements, each of them belongs either to

*A*, or to

*B*. So,

*å*

*À*

*Â*,

*if and only if*

*either*

*å*

*À*,

*or*

*å*

*Â*.

**
A product (
intersection
) of sets
**

*À*and

*Â*( it's written as

*À*

*Â*, Fig.2 ) is a set of elements, each of them belongs both to

*À*and to

*Â*. So,

*å*

*À*

*Â*,

*if and only if*

*å*

*À*

*and*

*å*

*Â*.

**
A difference of sets
**

*À*and

*Â*( it's written as

*À*–

*Â*, Fig.3 ) is a set of elements, which belong to the set

*A*, but don't belong to the set

*Â*. This set is called also a

**of the set**

*complement**B*relatively the set

*A*.

**
A symmetric difference of sets
**

*A*and

*B*

**( it's written as**

*À*\

*Â*), is called a set:

*
À
*
\
*
Â
*
= (
*
A
*
–
*
B
*
)
(
*
Â
*
–
*
A
*
) .

**
Properties of operations with
sets:
**

E x a m p l e s. 1. A set of children is a subset of the whole population.

2. An intersection of the set of integers and the set of positive

numbers is the set of natural numbers.

3. A union of the set of rational numbers and the set of irrational

numbers is the set of real numbers.

4. Zero is a complement of the set of natural numbers relatively

the set of non-negative integers.