# how to simplify and find equivalent of these equivalence formulas?

My question is about propositional logic.

Firstly:

How can i simplify the formula (F≡¬F) . In my opinion this is simply false ⊥), but i'm not sure about it.

Secondly :

For the formula (p≡q) , what is the equivalent formula that contains no connectives other than ⊃ (implication) and ⊥ (false) ?

Please help me! I could not find any resource on that. Thank you !

Yes :

$\lnot \bot \equiv \top$

and thus : $(\bot \equiv \lnot \bot) \equiv \bot$.

For $p \equiv q$, we have that it is equivalent to : $(p \to q) \land (q \to p)$.

But : $(p \land q)$ is $\lnot (p \to \lnot q)$, i.e.

$(p \to (q \to \bot)) \to \bot$.

• So, the answer is ((p⊃q)⊃((q⊃p)⊃⊥))⊃⊥ , right ? – user154335 Oct 31 '15 at 20:25
• @user154335 - yes (we can check it with truth-table). – Mauro ALLEGRANZA Oct 31 '15 at 20:27
• Adding that, how can i convert (p⊃q) and (p^q) to its equivalent formula that contains no connectives other than and v ? – user154335 Oct 31 '15 at 20:28
• thank you a lot ! For my question on comment, I convert (p⊃q) expression with disjunction, but i have nothing to eleminate negation in this expression. It seems unsolvable ! – user154335 Oct 31 '15 at 20:32
• @user154335 - probably because $\equiv$ and $\lor$ are not an adequate set of cnnecetives, and thus cannot express $\lnot$ ... – Mauro ALLEGRANZA Oct 31 '15 at 21:09