Question

Consider the following propositional statements:
$P_1:((A\wedge B)\to C))\equiv ((A\to C)\wedge (B\to C))$
$P_2:((A\vee B)\to C))\equiv ((A\to C)\vee (B\to C))$

Which one of the following is true?

• A. $P_1$ is a tautology , but not $P_2$
• B. $P_2$ is a tautology , but not $P_1$
• C. $P_1$ and $P_2$ are both tautologies
• D. $P_1$ and $P_2$ are not tautologies

Answer 1

The correct option is D $P_1$ and $P_2$ are not tautologies

(d)
$P_1:\left(\right(A\wedge B\to C\left)\right)\equiv \left(\right(A\to C)\wedge (B\to C\left)\right)$
LHS:
$(A\wedge B)\to$ C
$\equiv AB\to C$
$\equiv \left(AB\right){}^{\prime }+C$
$\equiv A{}^{\prime }+B{}^{\prime }+C$
RHS:
$(A\to C)\wedge (A\to C)$
$\equiv (A{}^{\prime }+C)(B{}^{\prime }+C)$
$\equiv A{}^{\prime }B{}^{\prime }+C$

Clearly ,LHS $\ne$ RHS
$P_1$ is not a tautology
$P_2:\left(\right(A\vee B\to C\left)\right)\equiv \left(\right(A\to C)\vee (B\to C)$
LHS $\equiv (A+B\to C)$
$\equiv (A+B){}^{\prime }+C)$
$\equiv A{}^{\prime }B{}^{\prime }+C$
RHS $\equiv (A\to C)\vee (B\to C)$
$\equiv (A{}^{\prime }+C)+(B{}^{\prime }+C)$
$\equiv A{}^{\prime }+B{}^{\prime }+C$
Clearly , LHS $\ne RHS\Rightarrow P_2$ is also not a tautology. Therefore , both $P_1$ and $P_2$ are not tautologies.
Correct choice is (d).