Question

To define the relation of logical equivalence on the set of all compound propositions is an equivalence relation.

Short answer

The equivalence class of, is the set of all tautologies and the equivalence class of, is the set of all contradictions.

Step-by-step solution

Given data

Given data is equivalence relation.

Concept used of equivalence relation

An equivalence relation is a binary relation that is reflexive, symmetric and transitive.

Solve for reflexive, symmetric and transitive

Two propositions are equivalent if their truth tables are identical. This relation is reflexive, since the truth table of a proposition is identical to itself.

It is symmetric, since if \(p\) and \(q\) have the same truth table, then q and \(p\) have the same truth table. There is one technical point about transitivity that should be noted.

We need to assume that the truth tables, as we consider them for three propositions p, q, and \(r\), have the same atomic variables in them.

Show the equivalence relation

If we make this assumption (and it cannot hurt to do so, since adding information about extra variables that do not appear in a pair of propositions does not change the truth value of the propositions), then we argue in the usual way: if \(p\) and \(q\) have identical truth tables, and if \(q\) and \(r\) have identical truth tables, then \(p\) and \(r\) have that same common truth table. The proposition \({\rm{T}}\) is always true; therefore the equivalence class for this proposition consists of all propositions that are always true, no matter what truth values the atomic variables have. Recall that we call such a proposition a tautology.

Therefore the equivalence class of . is the set of all tautologies. Similarly. the equivalence class of, is the set of all contradictions.