Solution: Assign a grade of A (correct), C (partially correct), or F (failure) to each

Chapter 2, Problem 18

(choose chapter or problem)

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justifyassignments of grades other than A. (a) Claim. For every indexed familyProof. Choose any Then by Theorem 2.3.2,Therefore, by transitivity of set inclusion, (b) Claim. If for all thenProof. Suppose Then, since for allTherefore, ax B. A B.x Aa B a ,aAa.aAa B a , Aa B.aAa aAa.aAa Ab and Ab aAa.Ab {Aa: a }.aAa a{Aa: a }, Aa.i=1Ai = .i=1Ai = {0, 1}.(c) Claim. For every indexed familyProof. LetThen aAa = {c, d} {a, b, c, d, e, f } =a{c, d, e, f }. Aa. = {r, s, t}, Ar = {a, b, c, d}, As = {b, c, d, e}, At =aAa a{Aa: a }, Aa.(d) Claim. For every indexed familyProof. Assume Then for someSince it is not the case that for someTherefore, for every But sincefor every This is a contradiction, so we conclude (e) Claim.Proof. Let Choose a natural number y such thatThus Therefore, x is an element ofSince for allTherefore, n=1[n, n + 1) = .n=1[n, n + 1) n , [n, n + 1) .