Question

Let the operations $\ast ,\odot \in \{\wedge ,\vee \}$. If $(p\ast q)\odot (p\odot \sim q)$ is a tautology, then the ordered pair $(\ast ,\odot )$ is

• A. $\vee ,\wedge$
• B. $\wedge ,\wedge$
• C. $\vee ,\vee$
• D. $\wedge ,\vee$

Answer 1

The correct option is C $\vee ,\vee$

$\ast ,\odot \in \{\wedge ,\vee \}$

Now for $(p\ast q)\odot (p\odot \sim q)$ is tautology

$\left(A\right)(\vee ,\wedge ):(p\vee q)\wedge (p\wedge \sim q)$ not a tautology

$\left(B\right)(\vee ,\vee ):(p\vee q)\vee (p\vee \sim q)$
$=P\vee T$ is tautology

$\left(C\right)(\wedge ,\wedge ):(p\wedge q)\wedge (p\wedge \sim q)$
$=(p\wedge p)\wedge (q\wedge \sim q)=p\vee F$ not a tautology (Fallasy)

$\left(D\right)(\wedge ,\vee ):(p\wedge q)\vee (p\vee \sim q)$ not a tautology