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Vp2Vp3) →q and apply a sequence of logical equivalences to obtain [(p₁ →q)^(p2 →q)^(p3 →q)]. !) Let r = p2 V p3, and show that (p1 V P2 V p3) →q is equivalent to (p₁ V r) → q. Note we really should only replace parenthesized subexpressions, so you should probably use the associativity of V to make (p2 V P3) appear as a subexpression first. 1) Apply the n = 2 version of the equivalence to your new expression from (i). You should obtain a compound expression with r→q as a subexpression. 1) Replace r with (p2 V p3), and apply the n = 2 equivalence again, this time to (p2 V P3) → q. ) Use associativity of A to show that your expression form (iii) is equivalent to the required form of [(P1 →q) ^ (P2 →q) ^ (P3 → q)].

Vp2Vp3) →q and apply a sequence of logical equivalences to obtain [(p₁ →q)^(p2 →q)^(p3 →q)]. !) Let r = p2 V p3, and show that (p1 V P2 V p3) →q is equivalent to (p₁ V r) → q. Note we really should only replace parenthesized subexpressions, so you should probably use the associativity of V to make (p2 V P3) appear as a subexpression first. 1) Apply the n = 2 version of the equivalence to your new expression from (i). You should obtain a compound expression with r→q as a subexpression. 1) Replace r with (p2 V p3), and apply the n = 2 equivalence again, this time to (p2 V P3) → q. ) Use associativity of A to show that your expression form (iii) is equivalent to the required form of [(P1 →q) ^ (P2 →q) ^ (P3 → q)].

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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discrete math 

Question B3: To justify case analysis, the text notes that the following expression is a tautology (section 1.8.2): [(P1 V P2 V ... V Pn) →q] → [(P₁ →q) ^ (P2 →q) ^ ^ (Pn →q)]. This is an infinite number of claims, asserting a different logical equivalence for each positive integer n. (P1 V P2)→q¬(P₁ VP2) V q = (P₁ ^ p2) V q = q V (P₁ ^ P2) = (qVp₁) ^ (q V p2) = (P₁ Vq) ^ ( P2 V q) = (P₁ →q) ^ (P2 → q)
Transcribed Image Text:Question B3: To justify case analysis, the text notes that the following expression is a tautology (section 1.8.2): [(P1 V P2 V ... V Pn) →q] → [(P₁ →q) ^ (P2 →q) ^ ^ (Pn →q)]. This is an infinite number of claims, asserting a different logical equivalence for each positive integer n. (P1 V P2)→q¬(P₁ VP2) V q = (P₁ ^ p2) V q = q V (P₁ ^ P2) = (qVp₁) ^ (q V p2) = (P₁ Vq) ^ ( P2 V q) = (P₁ →q) ^ (P2 → q)
(b) To verify the n = 3 equivalence (without building a 16 row truth table), we need to start with (p1 Vp2Vp3) →q and apply a sequence of logical equivalences to obtain [(p₁ →q)^(p2 →q)^(p3 →q)]. (i) Let r=p2 V p3, and show that (p₁ V p2 V P3) →q is equivalent to (p₁ Vr) →q. Note we really should only replace parenthesized subexpressions, so you should probably use the associativity of V to make (p2 V P3) appear as a subexpression first. (ii) Apply the n = 2 version of the equivalence to your new expression from (i). You should obtain a compound expression with r→q as a subexpression. (iii) Replace r with (p2 V P3), and apply the n = 2 equivalence again, this time to (p2 V P3) →q. (iv) Use associativity of A to show that your expression form (iii) is equivalent to the required form of [(P1 →q) ^ (P2 →q) ^ (P3 → q)]. (c) How could you extend the approach in (b) to verify the n = 4 version of the tautology? What if you needed a case analysis involving 5 or 742 cases?
Transcribed Image Text:(b) To verify the n = 3 equivalence (without building a 16 row truth table), we need to start with (p1 Vp2Vp3) →q and apply a sequence of logical equivalences to obtain [(p₁ →q)^(p2 →q)^(p3 →q)]. (i) Let r=p2 V p3, and show that (p₁ V p2 V P3) →q is equivalent to (p₁ Vr) →q. Note we really should only replace parenthesized subexpressions, so you should probably use the associativity of V to make (p2 V P3) appear as a subexpression first. (ii) Apply the n = 2 version of the equivalence to your new expression from (i). You should obtain a compound expression with r→q as a subexpression. (iii) Replace r with (p2 V P3), and apply the n = 2 equivalence again, this time to (p2 V P3) →q. (iv) Use associativity of A to show that your expression form (iii) is equivalent to the required form of [(P1 →q) ^ (P2 →q) ^ (P3 → q)]. (c) How could you extend the approach in (b) to verify the n = 4 version of the tautology? What if you needed a case analysis involving 5 or 742 cases?
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