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Use the laws of propositional logic to prove that the following compound propositions are logically equivalent. a. (p V q) ^ ¬(¬p ^ q) and p b. (p → r)^ (q →r) and (p V q) → r c. ¬((p ^ (q → r)) V r) and (p → q) ^ ¬r d. p+ (p ^q) and p → q

Use the laws of propositional logic to prove that the following compound propositions are logically equivalent. a. (p V q) ^ ¬(¬p ^ q) and p b. (p → r)^ (q →r) and (p V q) → r c. ¬((p ^ (q → r)) V r) and (p → q) ^ ¬r d. p+ (p ^q) and p → 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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Solve part a and b plz by using laws.

4. Use the laws of propositional logic to prove that the following compound propositions are logically equivalent. a. (p V q) A ¬(¬p ^ q) and p b. (p → r) A (q →r) and (p V q) →r c. ¬((p^ (q → r)) V r) and (p → q) ^ ¬r d. pe (рлq)and p - q
Transcribed Image Text:4. Use the laws of propositional logic to prove that the following compound propositions are logically equivalent. a. (p V q) A ¬(¬p ^ q) and p b. (p → r) A (q →r) and (p V q) →r c. ¬((p^ (q → r)) V r) and (p → q) ^ ¬r d. pe (рлq)and p - q
Idempotent laws p V p = p d = dy d (p ^ q) Ar = p ^ (q ^r) Associative laws ( φν) Vr=pV (q Vr . Commutative laws p V q = q V p p^ q = q ^p Distributive laws p V (q Ar) = (p V q) ^ (p V r) p^ (ą v r) = (p ^ q) v (p ^r) Identity laws pVF = p p^T = p Domination laws p V T = T p A F = F Double negation laws p קבר Complement (or negation) laws p V ¬p = T p^ ¬p = F De Morgan's laws ¬(p V q) = ¬p ^ ¬q ¬(p ^ q) = ¬p V ¬q Absorption laws pV (pΛq) p d = (b ^ d) v d Conditional identities p → q = -p V q реда(р-д) л (q — р)
Transcribed Image Text:Idempotent laws p V p = p d = dy d (p ^ q) Ar = p ^ (q ^r) Associative laws ( φν) Vr=pV (q Vr . Commutative laws p V q = q V p p^ q = q ^p Distributive laws p V (q Ar) = (p V q) ^ (p V r) p^ (ą v r) = (p ^ q) v (p ^r) Identity laws pVF = p p^T = p Domination laws p V T = T p A F = F Double negation laws p קבר Complement (or negation) laws p V ¬p = T p^ ¬p = F De Morgan's laws ¬(p V q) = ¬p ^ ¬q ¬(p ^ q) = ¬p V ¬q Absorption laws pV (pΛq) p d = (b ^ d) v d Conditional identities p → q = -p V q реда(р-д) л (q — р)
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