I need to prove/disprove the logical equivalences of the following two statements using basic equivalences:
$p \to (q \to r)$ and $(p \to q) \to r$.
$p \to (q \to r)$ and $q \to (p \to r)$.
I also need to verify each of the following by writing an equivalence proof:
$p \to (q \wedge r) = (p \to q) \wedge (p \to r)$.
$(p \to q) \wedge (p \vee q) = q$.
Also, I have two other questions:
Suppose $a$ and $b$ are integers and $a^2 - 5b$ is even. Prove that $b^2 - 5a$ is even.
Prove that for all integers $n$, $n \ge 1$: $1 + 3 + 5 + \cdots + (2n -1) = n^2$.
Thank you if you can help, I can do everything else on my work except these damn proofs!