Question

Help with propositional logic

Simplify
[~(p→q)→~(q→p)]∧(p∨q)

Where:
~ means negation
∧ means conjuction
∨ means disjuction
→ is the condicional.

Answer 1

Since:
\[
[\neg(p\to q)\to\neg (q\to p)]
\]Is
\[
[(q\to p)\to (p\to q)]
\]Which is
\[
[q\wedge p\to (p\to q)]
\]This reduces to:
\[
[q\wedge p \wedge p \to q]
\]Or, simply:
\[
[q\wedge p]
\]So, this is the simplest form (we can ignore the last term).

Answer 2

Does that make sense, @Juarismi ?

Answer 3

Shouldn't be p∨q ?
How do you go from
[(q→p)→(p→q)] to
[q∧p→(p→q)]

Is there a rule or something?

Answer 4

Crap, wait, you're right, I screwed up.
\[
[(\neg p\vee q)\to (p\to q)]
\]Which is:
\[
[((\neg p\vee q)\wedge p) \to q]
\]Or:
\[
(\neg p \wedge p) \vee (p\wedge q)\wedge p\to q
\]And, we still get:
\[
p\wedge q
\]

Answer 5

ok, ok i got it, thanks.

Answer 6

Sure thing, mate