(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have to prove that ##(p \equiv q) \equiv ((p ∧ q) ∨ (¬p ∧ ¬q))##

With no premisses

In order to prove this, I first need to prove that:

##(p \equiv q) \to ((p ∧ q) ∨ (¬p ∧ ¬q))##

And:

##((p ∧ q) ∨ (¬p ∧ ¬q)) \to (p \equiv q)##

I was able to find the second implication, but I am still looking how I can prove the first one.

2. Relevant equations

I have the inference rules, contraposition, modus tollens ...

3. The attempt at a solution

I started with the hypothetical statement:

##(p \equiv q)##

But then I need to end the hypothetical statement with:

##((p ∧ q) ∨ (¬p ∧ ¬q))##

So I have tried to start a new hypothetical statement:

##¬((p ∧ q) ∨ (¬p ∧ ¬q))##

And to introduce a negation.

##¬((p ∧ q) ∨ (¬p ∧ ¬q)) \to (p \equiv q)## was easy to find, but I can't find a way to prove:

##¬((p ∧ q) ∨ (¬p ∧ ¬q)) \to ¬(p \equiv q)##

So that I could eliminate the negation afterwards.

I was looking to prove

##¬(p \equiv q)##

But I couldn't find a way to introduce the negation.

So I've tried to eliminate the disjuntion of:

##(p ∧ q) ∨ (¬p ∧ ¬q)##

By starting a new hypothetical statement, but this couldn't help me any futher.

Thank you in advance!

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# Homework Help: Question about propositional logic

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