Can anyone please help with a Logic?

Question

I'm a bit confused on how to use the indirect proof.
The problem is: "=" : BIconditional and

~(p≡q) conditional [(p conditional q) conditional ~ (q conditional )]

THANKS

Answer 1

¬(p↔q) → ((p→q) → ¬(q→p))

This is easy to see intuitively. p↔q means simply that both p→q and q→p, so if we let s=(p→q) and t=(q→p), this translates to:

¬(s∧t) → (s → ¬t).

And that is of course true, since for (s → ¬t) to be false, s would have to be true and ¬t false, which means s is true t is true, so s∧t is true. Since, if ¬(s∧t) is true, s∧t is false, it follows that (s → ¬t) cannot be false, and thus is true. Q.E.D.

Of course, we can verify this formally using truth tables (I've arranged this one vertically, since the usual horizontal format is too wide for this post to contain):

p ............. ...................... ................... T T F F
q ............. ...................... ................... T F T F
p→q ....................... ................... ....... T F T T
q→p ....................... ................... ....... T T F T
¬(q→p) ......................... ..................... F F T F
(p→q)→¬(q→p) ................... ............... F T T F
p↔q ............................. ..................... T F F T
¬(p↔q) ........................ ...................... F T T F
¬(p↔q) → ((p→q)→¬(q→p)) ................. T T T T

Edit: fixed the spacing on the truth table.