functional completeness of $\{\to\}$

Question

Given that the set {∨, $\wedge$ , ¬} is functionally complete, how would I prove whether the set $\{\to\}$ is functionally complete?

expressing $→$ in terms of $∨$: $¬A∨B$

expressing $→$ in terms of $∧$: $¬(A∧¬B)$

I understand the above two expressions, but cannot seem to prove that it is/is not functionally complete.

Answer 1

Suppose I have an expression $\pi$ involving propositional variables $A_1, . . . , A_n$, but only the connective "$\implies$" (OK, fine, and parentheses).

• What can I say about the truth value of $\pi$ if I assign "True" to each of the $A_1, . . . , A_n$?

• What does that tell you about whether $\{\implies\}$ is functionally complete?