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I have an assignment and I need to prove the following logical equivalence using Laws of Logic and not using Truth Table:

p → q ≡ ~q → ~p

LAWS OF LOGIC:

1.Commutative Law: p ↔ q ≡ q ↔ p

2.Implication Laws: p →q ≡ ~p ∨ q ≡ ~(p ∧ ~q)

3.Exportation Law: (p ∧ q)→r ≡ p →(q →r)

4.Equivalence: p ↔ q ≡ (p →q)∧(q →p)

5.Reductio ad absurdum p →q ≡ (p ∧ ~q) →c

Kindly help me solving it, need to present it in 24 hours. Thanks all

Khubaib Khawar
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  • This wiki may help:https://proofwiki.org/wiki/Rule_of_Transposition/Formulation_1 – Brian Cheung May 12 '16 at 16:33
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    Use Implication law: $p \to q \equiv \lnot p \lor q$ and $\lnot q \to \lnot p \equiv \lnot \lnot q \lor \lnot p$. – Mauro ALLEGRANZA May 12 '16 at 16:48
  • If we use Implication Law the we will get p → q ≡ ¬p ∨ q

    and

    ¬q → ¬p ≡ ¬¬q ∨ ¬p

    Using Double Negation we get

    ¬q → ¬p ≡ q ∨ ¬p

    so would it be equal to the answer of L.H.S?

     – Khubaib Khawar May 12 '16 at 16:51

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