Which of the following statements is logicall | vedclass

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(D) The biconditional statement $p \leftrightarrow q$ is equivalent to $(p 	o q) \land (q 	o p)$.Thus, $∼ (p \leftrightarrow q) \equiv ∼ ((p 	o q)

Vedclass · Accepted Answer

$p \leftrightarrow ∼ q$

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(D) The biconditional statement $p \leftrightarrow q$ is equivalent to $(p 	o q) \land (q 	o p)$.Thus, $∼ (p \leftrightarrow q) \equiv ∼ ((p 	o q) \land (q 	o p))$.Using De Morgan's Law, this becomes $∼ (p 	o q) \lor ∼ (q 	o p)$.Since $p 	o q \equiv ∼ p \lor q$, then $∼ (p 	o q) \equiv p \land ∼ q$.Similarly, $∼ (q 	o p) \equiv q \land ∼ p$.So, $∼ (p \leftrightarrow q) \equiv (p \land ∼ q) \lor (q \land ∼ p)$.Now, check option $(D)$: $p \leftrightarrow ∼ q \equiv (p 	o ∼ q) \land (∼ q 	o p) \equiv (∼ p \lor ∼ q) \land (q \lor p) \equiv (p \land q) \lor (∼ p \land ∼ q)$. This is not it.Wait, let us check the truth table for $p \leftrightarrow ∼ q$:If $p=T, q=T$, then $p \leftrightarrow ∼ q$ is $T \leftrightarrow F = F$. $∼ (p \leftrightarrow q)$ is $∼ (T \leftrightarrow T) = F$.If $p=T, q=F$, then $p \leftrightarrow ∼ q$ is $T \leftrightarrow T = T$. $∼ (p \leftrightarrow q)$ is $∼ (T \leftrightarrow F) = T$.If $p=F, q=T$, then $p \leftrightarrow ∼ q$ is $F \leftrightarrow F = T$. $∼ (p \leftrightarrow q)$ is $∼ (F \leftrightarrow T) = T$.If $p=F, q=F$, then $p \leftrightarrow ∼ q$ is $F \leftrightarrow T = F$. $∼ (p \leftrightarrow q)$ is $∼ (F \leftrightarrow F) = F$.The truth values match for all cases. Thus, $∼ (p \leftrightarrow q) \equiv p \leftrightarrow ∼ q$.

Which of the following statements is logically equivalent to $∼ (p \leftrightarrow q)$?

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