Problem

Show that neither of the following two formulas tautologically imply each other \begin{gather}(A\leftrightarrow(B\leftrightarrow C))\ ((A\land(B\land C))\lor((\lnot A)\land((\lnot B)\land (\lnot C))))\end{gather}

Solutions

First, I show that the first formula does not tautologically imply the second. Consider the truth assignment \(v\) where \(v(A) = T\), \(v(B) = F\), and \(v(C)=F\). Then \(\overline{v}\) satisfies the first formula, but does not satisfy the second.

Now, to show that the second formula does not tautologically imply the first, redefine the truth assignment \(v\) so that \(v(A)=v(B)=v(C)=F\). Then \(\overline{v}\) satisfies the second formula, but not the first.

Therefore, neither formula tautologically implies the other