Topic: dis. math question

Who knows about Truth Tables, Logic, and Proofs.
How to proof expressions shown below, using Equivalent Statements and not applying truth tables directly.

p ? ?(p ? s) ? (?s ? p)
?(p ? q) ? (p ? ?q) ? (q ? ?p)
-- Discrete Mathematics with Combinatorics
James A. Anderson, part 1.3, task 4

thanks in advance

Re: dis. math question

The rule names are taken from http://en.wikipedia.org/wiki/Propositional_logic

   ?(p ? s) ? (?s ? p)   
= (p ? s) ? (p ? ?s)    (Material implication)
= p ? (s ? ?s)     (Distribution (1))
= p ? TRUE     ( (s ? ?s) is a tautology)
= p


   ?(p ? q)
= ?((p ? q) ? (q ? p))     (Material Equivalence (1))
= ?((?p ? q) ? (?q ? p))     (Material Implication)
= ?(?p ? q) ? ?(?q ? p)     (De Morgan's Theorem (1))
= (p ? ?q) ? (q ? ?p)     (De Morgan's Theorem (2))

The whole thing is simply a matter of starting with the complex half of the equation, then applying rules that fit...

Re: dis. math question

For: "?(p ? q) ? (p ? ?q) ? (q ? ?p)"  haven't you only shown that "~(p ? q) ?  (p ? ?q) ? (q ? ?p)" and not the additional "(p ? ?q) ? (q ? ?p) ? ~(p ? q)" for the biconditional?

Re: dis. math question

All rules that I used are applicable in reverse too I believe? Or is there a step that you think doesn't work in reverse?

Re: dis. math question

Yeah, you're correct. Sorry.  I'm used to doing propositional logic with trees.