An adequate set of connectives for propositional logic is a set such that for every formula of propositional logic there is an equivalent formula with only connectives from that set. For example, the set {, ∨} is adequate for propositional logic, because any occurrence of ∧and →can be removed by using the equivalences ϕ→ψ≡ϕ∨ψand ϕ∧ψ≡(ϕ∨ψ). (a) Show that {, ∧},{, →} and {→, ⊥} are adequate sets of connectives for propositional logic. (In the latter case, we are treating ⊥as a nullary connective.)
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