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TABLE I Set Identities.

: (A): A

A n @ . c ) : ( A n B ) n c

A n ( B u c ) : ( A n B ) u ( A n c ) A u ( B n c ) : ( A u B ) n ( A u c )

A n B : E v E

A U ( A n B ) : A A n ( A u B ) : A

EXAMPLEg Let 1:la.-e,i,o,uj (where the universal set is the set

• f letters
• f

bet). Then A : {b, c, d,

• f,

g, h, j, k, l, m, k, p, e, r, s, t, y, w, x, y, z}. the English al EXAMPTE 9 Let A be the set

• f positive

integers greater than l0 (with universal set the set

• f all

integers). Then [: {1,2,3,4,5,6,7,8,9, 10}.

Set Identities

Table I lists the most important set identities. We will prove several

• f these

identities using three different methods. These methods are presented to illustrate that there are many different approaches to the solution

• f a

problem. The proofs

• f the

remaining idenl will be left as exercises. The reader should note the similarity between these set identities the logical equivalences discussed in Section l.2.Infact, the set identities given can be directly from the corresponding logical equivalences. Furthermore, both ire special c identities that hold for Boolean algebra (discussed in Chapter 11). One way to show that two sets are equal is to show that each is a subset

• f the
• ther.

that to show that

• ne

set is a subset

• f a second

set, we can show that if an element the first set, then it must also belong to the second set. We generally use a direct proof to do we illustrate this type

• f proof by establishing

the second

• f De Morgan3

laws.

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