Page 168 - HOW TO PROVE IT: A Structured Approach, Second Edition

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154 Proofs
(b) Prove that there is a unique A ∈ P (U) such that for every B ∈
P (U), A ∪ B = A.
p d7. Let U be any set.
(a) Prove that there is a unique A ∈ P (U) such that for every B ∈
P (U), A ∩ B = B.
(b) Prove that there is a unique A ∈ P (U) such that for every B ∈
P (U), A ∩ B = A.
p d8. Let U be any set.
(a) Prove that for every A ∈ P (U) there is a unique B ∈ P (U) such
that for every C ∈ P (U), C \ A = C ∩ B.
(b) Prove that for every A ∈ P (U) there is a unique B ∈ P (U) such
that for every C ∈ P (U), C ∩ A = C \ B.
p d9. Recall that you showed in exercise 12 of Section 1.4 that symmet-
ric difference is associative; in other words, for all sets A, B, and C,
A (B C) = (A B) C. You may also find it useful in this prob-
lem to note that symmetric difference is clearly commutative; in other
words, for all sets A and B, A B = B A.
(a) Prove that there is a unique identity element for symmetric differ-
ence. In other words, there is a unique set X such that for every set
A, A X = A.
(b) Prove that every set has a unique inverse for the operation of symme-
tric difference. In other words, for every set A there is a unique set B
such that A B = X, where X is the identity element from part (a).
(c) Prove that for any sets A and B there is a unique set C such that
A C = B.
(d) Prove that for every set A there is a unique set B ⊆ A such that for
every set C ⊆ A, B C = A \ C.
p d10. Suppose A is a set, and for every family of sets F,if ∪F = A then
A ∈ F. Prove that A has exactly one element. (Hint: For both the exis-
tence and uniqueness parts of the proof, try proof by contradiction.)
p ∗
d 11. Suppose F is a family of sets that has the property that for every G ⊆
F, ∪G ∈ F. Prove that there is a unique set A such that A ∈ F and
∀B ∈ F(B ⊆ A).
12. (a) Suppose P(x) is a statement with a free variable x. Find a formula,
using the logical symbols we have studied, that means “there are
exactly two values of x for which P(x) is true.”
(b) Based on your answer to part (a), design a proof strategy for proving
a statement of the form “there are exactly two values of x for which
P(x) is true.”
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(c) Prove that there are exactly two solutions to the equation x = x .