For any formula, there exists an equivalent formula that contains no connectives other than ⊃ and ⊥. In this sense, {⊃, ⊥} is an “adequate” set of connectives.
I want to prove that {∧, ¬} is adequate.
My solution is, A and B is an atom. Recall that any general form (AVB) is truth-functionality equivalent to ¬(¬A∧¬B)
I prove this with truth table.
A|B|AVB|¬(¬A∧¬B)
t t| t | t
t f| t | t
f t| t | t
f f| f | f
yes it is adequate, but i want to prove it more general ex, any formula can prove this.
How can i prove it for general?
If you already know that $\{ ⊃, ⊥ \}$ is adequate, the simplest way to prove that also $\{ ∧, ¬ \}$ is so is to show how to define :
in terms of $\{ ∧, ¬ \}$.