Question

The statement formula:

$(A\to B)\to A)\leftrightarrow (B\to (B\to A))is$

• A. none of these
• B. satisfiable but not valid
• C. valid
• D. not satisfiable

Answer 1

The correct option is B satisfiable but not valid

option (a)

$LetS=\left(\right(A\to B)\to A)\leftrightarrow (B\to (B\to A\left)\right)$

Case 1 : When A is T.

$S\iff \left(\right(T\to B)\to T)\leftrightarrow (B\to (B\to T\left)\right)$

$\iff T\leftrightarrow T$

$\iff T$

Case 2 : When A is F.

$S\iff \left(\right(F\to B)\to F)\leftrightarrow (B\to (B\to F\left)\right)$

$\iff F\leftrightarrow \sim B$

$\iff B$

Which is a contingency.

$\therefore$ The given statement is satisfiable but not valid.