Simplifying Logical Expressions with Tautologies

Prove that a given relation $R$ on the set of integers $\mathbb\Z$ is reflexive, symmetric, and transitive.

For all x in the domain D, the predicate P(x) is true.

Determine if a proposition is a tautology, contradiction, or contingency using logical equivalences.

Verify the logical equivalence $\sim (\sim p \land q) \land (p \lor q) = p$ using the laws of logic.