To solve this problem, let's first understand the meaning of each logical operator:
1. [tex]\(p \vee q\)[/tex]: This represents the logical "or". It is true if either [tex]\(p\)[/tex] or [tex]\(q\)[/tex] is true (or both are true).
2. [tex]\(p \wedge q\)[/tex]: This represents the logical "and". It is true if both [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are true simultaneously.
3. [tex]\(p \rightarrow q\)[/tex]: This represents the logical "implication". It means "if [tex]\(p\)[/tex] then [tex]\(q\)[/tex]", and is true in all cases except when [tex]\(p\)[/tex] is true and [tex]\(q\)[/tex] is false.
Given:
- [tex]\(p\)[/tex]: [tex]\(x=4\)[/tex]
- [tex]\(q\)[/tex]: [tex]\(y=-2\)[/tex]
We need to represent the statement "If [tex]\(x=4\)[/tex], then [tex]\(y=-2\)[/tex]". This statement is a classical implication in logic, where the antecedent (if part) is [tex]\(p\)[/tex] and the consequent (then part) is [tex]\(q\)[/tex].
The appropriate logical representation for the statement "If [tex]\(x=4\)[/tex], then [tex]\(y=-2\)[/tex]" is [tex]\(p \rightarrow q\)[/tex].
Thus, the correct answer is:
[tex]\(p \rightarrow q\)[/tex]
So among the given options, the answer is the third option.
