fergznani
Answered

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Given:
[tex]\( p \)[/tex]: The zong is in the zung.
[tex]\( q \)[/tex]: The zong is not in the zam.

Which of the following is logically equivalent to [tex]\( p \rightarrow q \)[/tex]?

A. If the zong is not in the zung, then the zong is in the zam.
B. If the zong is not in the zam, then the zong is in the zung.
C. If the zong is in the zung, then the zong is in the zam.
D. If the zong is in the zam, then the zong is not in the zung.

Sagot :

GinnyAnswer
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.
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