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 inclusion. The inclusions
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
complement of the set 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 nonnegative integers.
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