Let A be a field of subsets of a non-empty set X
Let Form be the set of propositional formulas over a fixed set V of propositional variables. Let us call an arbitrary function $ν : V −→ A$ a valuation in A. By induction on formulas ν can be uniquely extended to a function $¯ν : Form −→ A$ such that for every α, β ∈ Form:
$¯ν(¬α) = X $ \ $ ν¯(α)$,
$¯ν(α ∨ β) = ¯ν(α) ∪ ν¯(β)$,
$¯ν(α ∧ β) = ¯ν(α) ∩ ν¯(β)$,
$¯ν(α ⇒ β) = X$ \ $(ν¯(α)) ∪ ν¯(β)$,
$¯ν(α ⇔ β) = X$ \ $(¯ν(α) - ν¯(β))$.
Prove that if $α ∈ Form$ is a tautology, then $¯ν(α) = X$.
I did the prove for tautology like $A ∨¬A$ but I dont know how to prove it for every tautology, because there re infinite tautologies. With the definition of tautology I just can say that for every valuation v, in fact in v, is true.