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I'm stuck in a discussion with a friend on some weekly tasks given to us and we're coming up with different opinions on this question that I'm hoping someone experienced might be able to expand on for us.

Firstly the question I have to prove Logical equivalence of using substitution and transitive property is;

a ↔ b ≡ (¬a ∨ b) ∧ (¬b ∨ a)

The answer I have come up with is;

a ↔ b ≡ ¬(a → ¬b) ∧ ¬(¬b → a)

a ↔ b ≡ (¬a ∨ b) ∧ (¬b ∨ a)

His answer is;

a → b ≡ ¬a ∨ b

a ↔ b ≡ (a → b ) ∧ (b → a)

a ↔ b ≡ (¬a ∨ b) ∧ (¬b ∨ a)

What's confusing me is in my friend's answer I thought we aren't allowed to change the left hand side of the equation, if we're proving a ↔ b, we can't add a → b in the left hand side? And that we should only be expanding on the right hand side, where we use transitivity and substitution to eventually prove the answer?

Any help would be greatly appreciated as I would love to learn what the correct answer is.

Bernard
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A G
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  • As you show, te two approaches give the same result. If you start with $(a \leftrightarrow b) \equiv (a \to b) \land (b \to a)$, the next step is to replace $(a \to b)$ with the equivalent $(\lnot a \lor b)$, and the same for the second conjunct. – Mauro ALLEGRANZA Apr 10 '19 at 07:41
  • See this post (and also this one) for the theorem regarding subst of equivalents. – Mauro ALLEGRANZA Apr 10 '19 at 07:44
  • ahh ok so he had it correct with a ↔ b ≡ (a → b ) ∧ (b → a)

    a ↔ b ≡ (¬a ∨ b) ∧ (¬b ∨ a)? But the a → b ≡ ¬a ∨ b was irrelevant?

     – A G Apr 10 '19 at 07:54
  • Not clear.... You want to prove that $(a \leftrightarrow b) \equiv (\text { something })$. Thus you have to start from $(a \leftrightarrow b)$ and replace it with equivalents. The only "rule" is : what is the list of equivalents you are using ? – Mauro ALLEGRANZA Apr 10 '19 at 07:57

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