Let $P$ be a group that it's elements are 257 sentences in which only atomic sentences from $A,B,C$ exist (i.e. $A \iff B,\space\space A \wedge B \wedge C, \space\space...$) Show that there exists two different $p_1, p_2 \in P$ so that the sentence $p_1 \iff p_2$ is a tautology.
Pigeons are the $257$ sentences but I can't think of a way to prove the asked.
On
Consider the truth tables. How many different possible truth tables are there for sentences with three variables? What can you say about sentences that have the same truth table?
On
How many different truth-functions are expressible using just three propositional variables? (Hint: how many lines are there in a truth-table in three variables? how many truth-possibilities are there for each line? So how many different truth-functions in three variables in total?)
If there are $x$ different truth-functions in three variables and $y$ different wffs built up from three variables, and $x < y$, how might the pigeon hole principle be applied?
HINT: Notice that $257=2^8+1$, so you might guess that there will be $2^8$ pigeonholes. You have three atomic sentences. They can have $2^3=8$ different combinations of truth values. How many different truth tables can you build from these $8$ combinations of truth values? Now notice that if $p_1\iff p_2$ is a tautology, then $p_1$ and $p_2$ have the same truth table. (Why?)