Proof through induction that all formulas with a certain characteristic are a tautology or logical equivalence of p

Question

First, sorry for the long title but I couldn't figure out how to summarize it better. This is a homework question for my course "Introduction to Logic" and I can't figure out how to solve it.

The question is:

Proof that each formula that only consists of the symbols $p$, $($, $)$, $\to$ is a tautology or a logical equivalent with $p$.

I first started with the base case.

Base Case: $(p -> p)$ is always a tautology.

Say $\phi = (p \to p)$ and $\psi$ is a arbitrary formula according to the rules mentioned above, than $(\psi \to \phi)$ is always a tautology and $(\phi \to \psi)$ is always a logical equivalent with $p$.

Is that something like a correct answer or do I miss something fundamental here? How can I do it right?

(Sorry if this question isn't asked right)

EDIT I tried to consider Jim's answer and came up with this. Is this right now? IMO it makes perfect sense ;-) I think the problem is that I don't know how to express it.

Statement: Each formula which consists only of the symbols p, (, ), $\to$ is a tautology or a logical equivalent with p.

Base case: $p$ is always a tautology and/or a logical equivalent with $p$

Induction step: Assume $(\psi \to \phi)$.

If $\phi$ is an arbitrary formula which is a tautology than $(\psi \to \phi)$ is always a tautology no matter what $\psi$ is.

If $\phi$ is an arbitrary formula which is a logical equivalent with $p$, than $(\psi \to \phi)$ is also a logical equivalent with p.

Therefore the statement is true.

Answer 1

First you've chosen the wrong base case, the base case you want is just the formula $p$. That formula is logically equivalent to $p$.

For your inductive step you can't choose $\phi$, you just have to assume it's an arbitrary formula which is either a tautology or equivalent to $p$. There are really two cases for the inductive step:

1. Something of the form $(Q)$ where $Q$ is either a tautology or equivalent to $p$.
2. Something of the form $Q \to R$ where each of $Q$ and $R$ is either a tautology or logically equivalent to $p$.

The first case is easy. For the second you have two choices for each of $Q$ and $R$ (tautology or equivalent to $p$) so there are four possible choices. Check for each of those choices that $Q \to R$ is either a tautology or logically equivalent to $p$.