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For example, let
A
=
{
1
,
3
,
5
,
7
,
9
}
and
B
=
{
3
,
5
,
6
,
10
,
11
}
. Then
A
∪
B
=
{
1
,
3
,
5
,
6
,
7
,
9
,
10
,
11
}
A
∩
B
=
{
3
,
5
}
A

B
=
{
1
,
7
,
9
}
A
±
B
=
{
1
,
7
,
9
,
6
,
10
,
11
}
It is often helpful to illustrate these combinations of sets using
Venn dia
grams
2
.
2.2.2
Properties of Operators
In this section, we investigate certain equalities between sets constructed
from our settheoretic operations.
P
ROPOSITION
2.6 (P
ROPERTIES OF OPERATORS
)
Let
A
,
B
and
C
be arbitrary sets. They satisfy the following properties:
Commutativity
Idempotence
A
∪
B
=
B
∪
A
A
∪
A
=
A
A
∩
B
=
B
∩
A
A
∩
A
=
A
Associativity
Empty set
A
∪
(
B
∪
C
) = (
A
∪
B
)
∪
C
A
∪ ∅
=
A
A
∩
(
B
∩
C
) = (
A
∩
B
)
∩
C
A
∩ ∅
=
∅
Distributivity
Absorption
A
∪
(
B
∩
C
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Unformatted text preview: ) = ( A ∪ B ) ∩ ( A ∪ C ) A ∪ ( A ∩ B ) = A A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A ∩ ( A ∪ B ) = A Proof We will just look at the ±rst distributivity equality. Some of the other cases will be set as exercises. Draw a Venn diagram to give some evidence that the property is indeed true. It is not a proof! Let A , B and C be 2 I will check whether you have been taught Venn diagrams. If not, we will go over them during lectures. 8...
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This note was uploaded on 01/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.
 Spring '09
 Koskesh
 Math, Sets

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