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Doubling the dimensions of a rectangle increases the area by a factor of 4. If [tex]\( p \)[/tex] represents doubling the dimensions of a rectangle and [tex]\( q \)[/tex] represents the area increasing by a factor of 4, which are true? Select two options.

A. [tex]\( p \rightarrow q \)[/tex] represents the original conditional statement.
B. [tex]\( \neg p \rightarrow \neg q \)[/tex] represents the inverse of the original conditional statement.
C. [tex]\( q \rightarrow p \)[/tex] represents the original conditional statement.
D. [tex]\( q \rightarrow p \)[/tex] represents the converse of the original conditional statement.
E. [tex]\( \neg q \rightarrow \neg p \)[/tex] represents the contrapositive of the original conditional statement.



Answer :

GinnyAnswer
Let's analyze the logical statements based on the given information.

Given:
- [tex]\( p \)[/tex] represents doubling the dimensions of a rectangle.
- [tex]\( q \)[/tex] represents the area increasing by a factor of 4.

Original conditional statement:
- [tex]\( p \rightarrow q \)[/tex]: If the dimensions of a rectangle are doubled, then the area increases by a factor of 4.

Inverse of the original conditional statement:
- [tex]\( \neg q \rightarrow \neg p \)[/tex]: If the area does not increase by a factor of 4, then the dimensions of the rectangle are not doubled.

Converse of the original conditional statement:
- [tex]\( q \rightarrow p \)[/tex]: If the area increases by a factor of 4, then the dimensions of the rectangle are doubled.

Contrapositive of the original conditional statement:
- [tex]\( \neg p \rightarrow \neg q \)[/tex]: If the dimensions of the rectangle are not doubled, then the area does not increase by a factor of 4.

Now, let’s identify the true statements based on logical equivalences:

1. [tex]\( p \rightarrow q \)[/tex] represents the original conditional statement.
2. [tex]\( \neg p \rightarrow \neg q \)[/tex] represents the contrapositive of the original conditional statement.

Both the original statement and its contrapositive are logically equivalent and true. Therefore, the two correct options are:

1. [tex]\( p \rightarrow q \)[/tex] represents the original conditional statement.
2. [tex]\( \neg p \rightarrow \neg q \)[/tex] represents the contrapositive of the original conditional statement.

So, the correct answers are:

1. [tex]\( p \rightarrow q \)[/tex]
2. [tex]\( \neg p \rightarrow \neg q \)[/tex]

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