Question

The following propositional statement is
$(P\to (Q\vee R\left)\right)\to \left(\right(P\wedge Q)\to R)$

• A. Satisfiable but not valid
• B. Valid
• C. A contraditiction
• D. None of the above

Answer 1

The correct option is A Satisfiable but not valid

$(P\to (Q\vee R\left)\right)\to \left(\right(P\wedge Q)\to R)$
$\equiv (P\to Q+R)\to (PQ\to R)$
$\equiv [P{}^{\prime }+Q+R]\to \left[\right(PQ){}^{\prime }+R]$
$\equiv [P{}^{\prime }+Q+R]\to [P{}^{\prime }+Q{}^{\prime }+R]$
$\equiv (P{}^{\prime }+Q+R){}^{\prime }+P{}^{\prime }+Q{}^{\prime }+R$
​​​​​​​$\equiv PQ{}^{\prime }R{}^{\prime }+P{}^{\prime }+Q{}^{\prime }+R$
​​​​​​​$\equiv Q{}^{\prime }+Q{}^{\prime }PR{}^{\prime }+P{}^{\prime }+R$
​​​​​​​$\equiv Q{}^{\prime }+P{}^{\prime }+R$ (by absorption law)
Which is a contingency (i.e. satisfiable but not valid).