Answer :
Let's break down the problem step-by-step to identify which expression represents the converse of a given conditional statement [tex]\( p \rightarrow q \)[/tex].
1. Understanding the Conditional Statement [tex]\( p \rightarrow q \)[/tex]:
- [tex]\( p \rightarrow q \)[/tex] means "if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]".
- In this context, [tex]\( p \)[/tex] is the hypothesis (or antecedent) and [tex]\( q \)[/tex] is the conclusion (or consequent).
2. Defining the Converse:
- The converse of a conditional statement [tex]\( p \rightarrow q \)[/tex] is obtained by swapping the hypothesis and the conclusion. Therefore, the converse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
3. Evaluating the Options:
- [tex]\(\sim p \rightarrow -q\)[/tex]: This expression states "if not [tex]\( p \)[/tex], then not [tex]\( q \)[/tex]". This does not swap the hypothesis and conclusion; it involves negations.
- [tex]\(\sim q \rightarrow \sim p\)[/tex]: This expression states "if not [tex]\( q \)[/tex], then not [tex]\( p \)[/tex]". This still does not represent the converse of the original statement.
- [tex]\(q \rightarrow p\)[/tex]: This expression correctly states "if [tex]\( q \)[/tex], then [tex]\( p \)[/tex]", which is precisely the converse of [tex]\( p \rightarrow q \)[/tex].
- [tex]\(p \rightarrow q\)[/tex]: This is the original conditional statement, not its converse.
From this detailed analysis, we can conclude that the correct answer, which represents the converse of the conditional statement [tex]\( p \rightarrow q \)[/tex], is:
[tex]\[ q \rightarrow p \][/tex]
1. Understanding the Conditional Statement [tex]\( p \rightarrow q \)[/tex]:
- [tex]\( p \rightarrow q \)[/tex] means "if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]".
- In this context, [tex]\( p \)[/tex] is the hypothesis (or antecedent) and [tex]\( q \)[/tex] is the conclusion (or consequent).
2. Defining the Converse:
- The converse of a conditional statement [tex]\( p \rightarrow q \)[/tex] is obtained by swapping the hypothesis and the conclusion. Therefore, the converse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
3. Evaluating the Options:
- [tex]\(\sim p \rightarrow -q\)[/tex]: This expression states "if not [tex]\( p \)[/tex], then not [tex]\( q \)[/tex]". This does not swap the hypothesis and conclusion; it involves negations.
- [tex]\(\sim q \rightarrow \sim p\)[/tex]: This expression states "if not [tex]\( q \)[/tex], then not [tex]\( p \)[/tex]". This still does not represent the converse of the original statement.
- [tex]\(q \rightarrow p\)[/tex]: This expression correctly states "if [tex]\( q \)[/tex], then [tex]\( p \)[/tex]", which is precisely the converse of [tex]\( p \rightarrow q \)[/tex].
- [tex]\(p \rightarrow q\)[/tex]: This is the original conditional statement, not its converse.
From this detailed analysis, we can conclude that the correct answer, which represents the converse of the conditional statement [tex]\( p \rightarrow q \)[/tex], is:
[tex]\[ q \rightarrow p \][/tex]
