**
Readings for Session 5 – (Continued)
**

*Properties of Union and Intersection of Sets*

The following set properties are
given here in preparation for the properties for addition and
multiplication in arithmetic. Note the close similarity between
these properties and their corresponding properties for addition
and multiplication.

**
Commutative
Properties:
**
The

General Properties:
*A** *
∪
*B* =
*B** *
∪
*A*
and
*A*
∩
*B* =
*B*
∩ *A.*

*
Example:*
Let
*A* = {*x*
: *x* is a whole number
between 4 and 8} and *B*
= {*x* :
*x* is an even natural
number less than 10}.

Then *A** *
∪
*B* = {5, 6, 7}* *
∪
{2, 4, 6, 8} = {2, 4, 5, 6, 7, 8} = {2, 4, 6, 8}* *
∪
{5, 6, 7} = *B** *
∪
*A
*and

**
Associative Properties:
**
The

General Property: (*A** *
∪
*B*)* *
∪
*C* =
*A** *
∪
(*B** *
∪
*C*) and (*A*
∩ *B*)
∩ *C* =
*A*
∩ (*B*
∩ *C*)

*Example:*
Let *A* = {a, n,
t}, *B* = {t, a, p},
and *C* = {s, a, p}.

Then*A** *
∪
*B*)* *
∪
*C* = {p, a, n, t}* *
∪
{s, a,
p} = {p, a, n, t, s} = {a, n, t}* *
∪
{t, a,
p, s} = *A** *
∪
(*B** *
∪ *
C*)

and
(*A*
∩ *B*)
∩ *C* = {a, t}
∩ {s, a,
p} = {a} = {a, n, t}
∩ {a, p}
= *A*
∩ (*B*
∩ *C*)

**
Identity
Property for Union: **
The

General Property:
*A *
∪
∅*
*
=
∅* *
∪
*A* =
*A*

*Example:*
Let *A* = {3, 7, 11}
and *B* = {*x*
: *x* is a natural
number less than 0}.

Then *A** *
∪
*
B*
= {3, 7, 11}* *
∪
{ } = {3, 7, 11}.

** Intersection
Property of the Empty Set:**
The

General Property:
*A*
∩
∅
=
∅
∩ *A* =
∅*.*

*Example:*
Let *A* = {3, 7, 11}
and *B* = {*x*
: *x* is a natural
number less than 0}.

Then *A*
∩ *B*
= {3, 7, 11}
∩ { } = { }.

What number has a similar property
when multiplying whole numbers? What is the corresponding
property for multiplication of whole numbers?

**
Distributive
Properties: **
The

General Property:
*A** *
∪
(*B*
∩ *C*) = (*A** *
∪
*B*)
∩ (*A** *
∪
*C*) and
*A*
∩ (*B** *
∪
*C*) = (*A*
∩ *B*)* *
∪
(*A*
∩ *C*)

*Example: *Let *A*
= {a, n, t}, *B* = {t,
a, p}, and *C* = {s, a,
p}. Then

*
A*

A

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