Use only logical equivalencies and material implication. Make sure to provide re
ID: 3143780 • Letter: U
Question
Use only logical equivalencies and material implication. Make
sure to provide reasoning for each step. Thank you in advance. The question is # 20.
Preview File EditView Go Tools Window Help tò 92%) Thu Aug 31 1:52 PM aE discrete-mathematics-and-its-applications-7th-edition-rosen.pdf (page 56 of 1.071) v Q Search V di Each of Exercises 16-28 asks you to show that two compound propositions are logically equivalent. To do this, either show that both sides are true, or that both sides are false, for exactly the same combinations of truth values of the propositional variables in these expressions (whichever is easier). 16. Show that p q and (p ^ q) v (p -y) are logically 40. Find a compound proposition involving the p variables p, q, and r that is true when p an and r is false, but is false otherwise. [Hint: junction of each propositional variable or its 41. Find a compound proposition involving the p variables p, q, and r that is true when exactly and r are true and is false otherwise. [Hint: junction of conjunctions. Include a conjunct combination of values for which the compou tion is true. Each conjunction should include three propositional variables or its negations rs' 42. Suppose that a truth table in n propositional specified. Show that a compound propositi truth table can be formed by taking the di conjunctions of the variables or their negatio equivalent. 17. Show that ,(p q) and p -y are logically equiva- lent. 18. Show that p q and-q -p are logically equivalent. 19. Show thatap q and p -y are logically equival 20. Show that-hpq) and p 21. Show that(p +4) and pq are logically equiva- 34 q are logically equivalent. 22. Show that (p q) (p-> r) and p (q ^ r) are log- 23. Show that (p r) ^ (q r) and (p v q) r are log- 24. Show that (p q) v (p r) and p (q v r) are log- 25. Show that (p r) V (q r) and (p ^ q) r are log- lent. ically equivalent. ically equivalent. ically equivalent. conjunction included for each combination which the compound proposition is true. compound proposition is said to be in disj mal form. A collection of logical operators is called functio plete if every compound proposition is logically e IJExplanation / Answer
We have ~(A B)
= ~((A ^ ~B) v (~A ^ B))
= ~(A ^ ~B) ^ ~(~A ^ B) by De-Morgan's law
= (~A v B) ^ (A v ~B) by De-Morgan's law
= (~A v B) ^ (~B v A) (commutative law)
= (A -> B) ^ (B -> A) by Material implication
= A <-> B
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