Answer :
To determine which of the expressions is equivalent to [tex]\( p \vee q \)[/tex] (which states "p or q"), let's analyze each of the given options in detail:
Option A: [tex]\( \neg p \rightarrow q \)[/tex]
To understand this, recall that the implication [tex]\( \neg p \rightarrow q \)[/tex] can be rewritten as [tex]\( \neg(\neg p) \vee q \)[/tex] using logical equivalences. Simplifying [tex]\( \neg(\neg p) \)[/tex] gives us [tex]\( p \vee q \)[/tex]. Therefore, [tex]\( \neg p \rightarrow q \)[/tex] is indeed equivalent to [tex]\( p \vee q \)[/tex].
Option B: [tex]\( \neg(\neg p \vee \neg q) \)[/tex]
This is the negation of the disjunction [tex]\( \neg p \vee \neg q \)[/tex]. According to De Morgan's laws, [tex]\( \neg(\neg p \vee \neg q) \)[/tex] is equivalent to [tex]\( p \wedge q \)[/tex] (the conjunction "p and q"), which is logically different from [tex]\( p \vee q \)[/tex].
Option C: [tex]\( \neg\left(p \wedge q\right) \)[/tex]
According to De Morgan's laws again, [tex]\( \neg(p \wedge q) \)[/tex] is equivalent to [tex]\( \neg p \vee \neg q \)[/tex]. This is clearly not the same as [tex]\( p \vee q \)[/tex].
Option D: [tex]\( \neg(p) \)[/tex]
This option is just the negation of [tex]\( p \)[/tex], which is unrelated to [tex]\( q \)[/tex]. Hence, it cannot be equivalent to [tex]\( p \vee q \)[/tex].
Given these analyses, the expression that is equivalent to [tex]\( p \vee q \)[/tex] is:
A. [tex]\( \neg p \rightarrow q \)[/tex]
Thus, the correct choice is Option A.
Option A: [tex]\( \neg p \rightarrow q \)[/tex]
To understand this, recall that the implication [tex]\( \neg p \rightarrow q \)[/tex] can be rewritten as [tex]\( \neg(\neg p) \vee q \)[/tex] using logical equivalences. Simplifying [tex]\( \neg(\neg p) \)[/tex] gives us [tex]\( p \vee q \)[/tex]. Therefore, [tex]\( \neg p \rightarrow q \)[/tex] is indeed equivalent to [tex]\( p \vee q \)[/tex].
Option B: [tex]\( \neg(\neg p \vee \neg q) \)[/tex]
This is the negation of the disjunction [tex]\( \neg p \vee \neg q \)[/tex]. According to De Morgan's laws, [tex]\( \neg(\neg p \vee \neg q) \)[/tex] is equivalent to [tex]\( p \wedge q \)[/tex] (the conjunction "p and q"), which is logically different from [tex]\( p \vee q \)[/tex].
Option C: [tex]\( \neg\left(p \wedge q\right) \)[/tex]
According to De Morgan's laws again, [tex]\( \neg(p \wedge q) \)[/tex] is equivalent to [tex]\( \neg p \vee \neg q \)[/tex]. This is clearly not the same as [tex]\( p \vee q \)[/tex].
Option D: [tex]\( \neg(p) \)[/tex]
This option is just the negation of [tex]\( p \)[/tex], which is unrelated to [tex]\( q \)[/tex]. Hence, it cannot be equivalent to [tex]\( p \vee q \)[/tex].
Given these analyses, the expression that is equivalent to [tex]\( p \vee q \)[/tex] is:
A. [tex]\( \neg p \rightarrow q \)[/tex]
Thus, the correct choice is Option A.
