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]
