Certainly! Let's solve the problem step-by-step using logical equivalences.
Given logical statements:
- [tex]\( p \)[/tex]: The zong is in the zung.
- [tex]\( q \)[/tex]: The zong is not in the zam.
We need to find the statement logically equivalent to [tex]\( p \rightarrow q \)[/tex] (if [tex]\( p \)[/tex] then [tex]\( q \)[/tex]).
First, recall the logical equivalence properties:
1. [tex]\( p \rightarrow q \)[/tex] is equivalent to [tex]\( \neg p \lor q \)[/tex] (Not [tex]\( p \)[/tex] or [tex]\( q \)[/tex]).
2. The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex] (if Not [tex]\( q \)[/tex] then Not [tex]\( p \)[/tex]).
Let's analyze each option logically:
1. If the zong is not in the zung, then the zong is in the zam.
This is [tex]\( \neg p \rightarrow \neg q \)[/tex].
2. If the zong is not in the zam, then the zong is in the zung.
This is [tex]\( \neg q \rightarrow p \)[/tex].
3. If the zong is in the zung, then the zong is in the zam.
This is [tex]\( p \rightarrow \neg q \)[/tex].
4. If the zong is in the zam, then the zong is not in the zung.
This is [tex]\( q \rightarrow \neg p \)[/tex].
To find the statement logically equivalent to [tex]\( p \rightarrow q \)[/tex], we consider the contrapositive:
- [tex]\( \neg q \rightarrow \neg p \)[/tex].
By analyzing the options:
- Option 2, [tex]\( \neg q \rightarrow p \)[/tex], represents the contrapositive form of [tex]\( p \rightarrow q \)[/tex].
Thus, the statement that is logically equivalent to [tex]\( p \rightarrow q \)[/tex] is:
If the zong is not in the zam, then the zong is in the zung.
So, the correct option is number 2.
