PubTeX output 2020.08.14:1558
(Theorem4.3.7(i))
(Theorem4.3.6(ii))
(Theorem4.3.7(i))
8. Redo Section 4.7, Example 4 using the algebra of sets.
Solution: Suppose and . . . . (I)
Then (Lemma 3.2.1(i))
((I), Sub=)
(Theorem 4.3.6(i))
(Theorem 4.3.7(ii))
(Theorem 4.3.7(v))
(Theorem 4.3.4(i))
(Theorem 4.2.2(iii))
(Theorem 4.3.7(i))
(Theorem 4.3.7(v))
(Theorem 4.3.4(i))
We thus have: and
(Lemma 3.2.3(ii))
4.9 Exercises
1. Suppose x is a set such that . Of which of the following sets is x a member, x a subset, x neither a
member nor a subset?
(a)
(b) x
(c)
(d)
(e)
(f)
2. Let be sets such that , , and .
(a) Express the set D using only the symbols and ,.
(b) Determine which of the following is true:
3. Prove that if and , then .
4. Prove that .
5. Use only the theorems in this chapter to prove the following:
(a)
(b)
(c)
(d)
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6. (a) Let for any set X. Prove that .
(b) Let for any set X. Prove that .
7. Prove that iff
8. Prove that
(a) .
(b) .
(c) .
(d) .
(e) .
9. Let be sets. Prove that:
(a) iff .
(b) iff .
(c) iff .
(d) iff .
10. Let and be sets. Prove that .
11. Prove or disprove the following:
(a) .
(b) .
(c) .
(d) .
(e) .
12. Prove that if , then there exists a set C such that and .
13. Prove that if and , then .
14. Prove that:
(a)
(b)
(c)
(d)
15. A set equation in one variable is an equation of the form , where is an unknown set and
and are set expressions formed using and on and some fixed sets . The
following procedure determines a condition under which the equation has a solution for , and all solutions
if the condition is satisfied.
Procedure:
begin
1. Given . Construct . (note: )
2. Convert into an equation of the form such that the set
expressions and are free of , as follows:
(a) repeatedly apply Theorem 4.3.7(v) to replace subexpressions of the form to until
no such expression exists;
(b) repeatedly apply Theorem 4.3.6 (DeMorgan’s Theorem) to subexpressions of the form until no
such application can be applied;
(c) repeatedly apply Theorem 4.3.7(i) to subexpressions of the form (note: or
) until no such subexpression exists;
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