Using the laws of logic (algebraic version) to show the following equivalences

Question

I have some questions about algebra and discrete, with using law of logic. I am not sure how to prove the equivalences.

Can someone please show me how this works and show the equivalence using the law of logic?

(b) (p → (q → r)) ≡ (q → (p → r))

(c) ((p → r) ∨ (q → r)) ≡ ((p ∧ q) → r)

Answer 1

You have to apply various identities such as DeMorgan's law. Take a look at http://www.csm.ornl.gov/~sheldon/ds/sec1.1.html

You can prove these identities with truth tables and then use the formal approach for efficiency. Let's do (a)

\begin{align} \overline{\overline p \land q} \equiv \overline{\overline p} \lor \overline q \equiv p \lor \overline q \equiv \overline p \rightarrow \overline q \equiv q \rightarrow p \end{align}

Where I have used, in order: DeMorgan's law for negation of a conjuction, double negation resolves to identity , then a -> b equivalent to not a or b and finally contrapositive.