Propositional Logic

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.)

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

Truth Tables and Logical Statements | IIT JEE Study Material

Truth Table | Definition, Examples & Rules