Prove that semantic equivalence is reflexive, symmetric and transitive.

Question

Show that the relation ≡ is

(a) reflexive: φ ≡ φ holds for all φ

(b) symmetric: φ ≡ ψ implies ψ ≡ φ and

(c) transitive: φ ≡ ψ and ψ ≡ η imply φ ≡ η.

This is simply practise for an upcoming Quiz in Predicate Logic. I have very little idea how to use the tools we've learned to set up a proof.

Answer 1

Using the definition "the truth tables of both formulae are the same" of semantic equivalence, translate (a), (b) and (c) into English sentences. For example: (a) reflexive: the truth table of $\varphi$ is the the same as the truth table of $\varphi$

You can see that this is obviously true. The others are as well.