MACM 101 Assignment II: Due February 6 at the beginning of class 1. Determine whether ∀x(P(x) ↔ Q(x)) and ∀xP(x) ↔ ∀xQ(x) are logically equivalent. Justify your answer. 2. Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). (a) ¬∃y∃xP(x, y) (b) ¬∀x∃yP(x, y) (c) ¬∃y(Q(y) ∧ ∀x¬R(x, y)) (d) ¬∃y(∃xR(x, y) ∨ ∀xS(x, y)) (e) ¬∃y(∀x∃zT(x, y, z) ∨ ∃x∀zU(x, y, z)) 3. Justify the rule of universal transitivity, which states that if ∀x(P(x) → Q(x)) and ∀x(Q(x) → R(x)) are true, then ∀x(P(x) → R(x)) is true, where the domains of all quantifiers are the same. 4. Use rules of inference to show that if ∀x(P(x) ∨ Q(x)) and ∀x((¬P(x) ∧ Q(x)) → R(x)) are true, then ∀x(¬R(x) → P(x)) is also true, where the domains of all quantifiers are the same. 5. Prove that if n is a perfect square, then n + 2 is not a perfect square. 6. Prove that if n is an integer and 3n + 2 is even, then n is even using (a) a proof by contraposition. (b) a proof by contradiction. 7. Show that if A ⊆ C and B ⊆ D, then A × B ⊆ C × D. 8. Let A, B, and C be sets. Show that (A − B) − C = (A − C) − (B − C). 9. Can you conclude that A = B if A, B, and C are sets such that (a) A ∪ C = B ∪ C? (b) A ∩ C = B ∩ C? (c) A ∪ C = B ∪ C and A ∩ C = B ∩ C? Justify your answer in each case. 1

AccountingQueen

(3)

Status NEW Posted 04 Sep 2017 12:09 PM My Price 8.00

Hel-----------lo -----------Sir-----------/Ma-----------dam----------- T-----------han-----------k y-----------ou -----------for----------- us-----------ing----------- ou-----------r w-----------ebs-----------ite----------- an-----------d a-----------cqu-----------isi-----------tio-----------n o-----------f m-----------y p-----------ost-----------ed -----------sol-----------uti-----------on.----------- Pl-----------eas-----------e p-----------ing----------- me----------- on----------- ch-----------at -----------I a-----------m o-----------nli-----------ne -----------or -----------inb-----------ox -----------me -----------a m-----------ess-----------age----------- I -----------wil-----------l

Not Rated(0)

• Buy Answer