epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Exercises Without Solutions 113
(This show that this operation is a symmetric (commutative), because
the roles of A and B can be changed without changing the final result.)
79. Let A, B, C, and D be any subsets of a universal set U. In each case,
either give a proof of the fact that the equahty is true or find a
counterexample to show that it is false. Do not use Venn diagrams to
prove the truth of equalities.
(A-B)nc = An(B'nc)
(AuBucy
= A'nB'nC
(AUB)U{CnD) = (AUBUC)nD
80. Let lun = ritrP • Then lim a„ = 0.
81. Let P{x) and Q{x) be two polynomials such that P(x) = (x^ + 1)6(^)-
The solution sets of the two polynomials coincide.
(Hint: Prove that XQ is a zero of P{x) if and only if it is a zero of Q{x).)
82. Let €i and I2 be two nonhorizontal distinct fines perpendicular to a
third fine £3. Then ti and £2 are parafiel to each other.
(Hint: work on the slopes of the three fines.)
83. There exists a unique prime of the form n^ - 1.
84. Let X be a real number. If x^ is not a rational number, then x is not a
rational number.
85. The sum of two odd functions is an odd function.