To identify the contrapositive of a conditional statement, let's start by understanding the given conditional statement [tex]\( p \rightarrow q \)[/tex]. This means "if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]".
A contrapositive reverses and negates both the hypothesis and the conclusion of the original statement. Here are the breakdowns of the options provided:
1. [tex]\( q \rightarrow p \)[/tex]: This is the converse of [tex]\( p \rightarrow q \)[/tex].
2. [tex]\( \sim q \rightarrow \sim p \)[/tex]: This is obtained by negating both the hypothesis and the conclusion of the original statement and then reversing them. Thus, "if not [tex]\( q \)[/tex], then not [tex]\( p \)[/tex]".
3. [tex]\( p \rightarrow q \)[/tex]: This is simply the original conditional statement.
4. [tex]\( \sim p \rightarrow \sim q \)[/tex]: This is the inverse of [tex]\( p \rightarrow q \)[/tex].
Therefore, the contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex]. The correct answer is:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
