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Which statement is logically equivalent to the following conditional statement?

"If it is a rectangle, then it does not have exactly three sides."

A. If it has exactly three sides, then it is not a rectangle.
B. If it is not a rectangle, then it has exactly three sides.
C. If it does not have exactly three sides, then it is a rectangle.
D. If it is not a rectangle, then it does not have exactly three sides.

Sagot :

GinnyAnsweridn
To determine which statement is logically equivalent to the given conditional statement, "If it is a rectangle, then it does not have exactly three sides," we need to consider the concept of logical equivalence in conditional statements.

In logic, a conditional statement "If P, then Q" (denoted [tex]\( P \rightarrow Q \)[/tex]) can be transformed into several equivalent forms:
1. The contrapositive: "If not Q, then not P" (denoted [tex]\( \neg Q \rightarrow \neg P \)[/tex]) which is logically equivalent to the original statement.
2. The inverse: "If not P, then not Q" (denoted [tex]\( \neg P \rightarrow \neg Q \)[/tex]).
3. The converse: "If Q, then P" (denoted [tex]\( Q \rightarrow P \)[/tex]).

Let's analyze each of the provided options for logical equivalence:

1. "If it has exactly three sides, then it is not a rectangle."
- In terms of our statement, this translates to: If Q is true, then P is false (contrapositive form [tex]\( \neg Q \rightarrow \neg P \)[/tex]).
- This matches the logical structure of the contrapositive of "If it is a rectangle, then it does not have exactly three sides" which is "If it has exactly three sides, then it is not a rectangle".

2. "If it is not a rectangle, then it has exactly three sides."
- This is the inverse form ([tex]\( \neg P \rightarrow \neg Q \)[/tex]).
- This is not logically equivalent to the original statement.

3. "If it does not have exactly three sides, then it is a rectangle."
- This is the converse form ([tex]\( Q \rightarrow P \)[/tex]).
- This is also not logically equivalent to the original statement.

4. "If it is not a rectangle, then it does not have exactly three sides."
- This is a different form of statement ([tex]\( \neg P \rightarrow \neg Q \)[/tex]).
- This is not logically equivalent to the given statement either.

Therefore, the statement that is logically equivalent to "If it is a rectangle, then it does not have exactly three sides" is:

- "If it has exactly three sides, then it is not a rectangle."

Hence, the correct choice is:

Choice 1: "If it has exactly three sides, then it is not a rectangle."
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