Answer :
Let's break down the problem and clarify each logical expression based on the given statement and the defined terms:
1. Original Conditional ( [tex]\(p \rightarrow q\)[/tex] ):
- Statement: If [tex]\(p\)[/tex] represents doubling the dimensions of a rectangle and [tex]\(q\)[/tex] represents the area increasing by a factor of 4, then [tex]\(p \rightarrow q\)[/tex] is stated as "If the dimensions of a rectangle are doubled, then the area increases by a factor of 4."
- Truth Value: True
2. Inverse ( [tex]\(-p \rightarrow -q\)[/tex] ):
- Statement: If [tex]\(-p\)[/tex] represents not doubling the dimensions of a rectangle and [tex]\(-q\)[/tex] represents the area not increasing by a factor of 4, then [tex]\(-p \rightarrow -q\)[/tex] is stated as "If the dimensions of a rectangle are not doubled, then the area does not increase by a factor of 4."
- Truth Value: False
3. Converse ( [tex]\(q \rightarrow p\)[/tex] ):
- Statement: If [tex]\(q\)[/tex] represents the area increasing by a factor of 4 and [tex]\(p\)[/tex] represents doubling the dimensions of a rectangle, then [tex]\(q \rightarrow p\)[/tex] is stated as "If the area increases by a factor of 4, then the dimensions of the rectangle are doubled."
- Truth Value: True
4. Contrapositive ( [tex]\(-q \rightarrow -p\)[/tex] ):
- Statement: If [tex]\(-q\)[/tex] represents the area not increasing by a factor of 4 and [tex]\(-p\)[/tex] represents not doubling the dimensions of a rectangle, then [tex]\(-q \rightarrow -p\)[/tex] is stated as "If the area does not increase by a factor of 4, then the dimensions of the rectangle are not doubled."
- Truth Value: False
Based on these logical evaluations, the two true statements are:
1. [tex]\(p \rightarrow q\)[/tex] ("If the dimensions of a rectangle are doubled, then the area increases by a factor of 4.") is true.
2. [tex]\(q \rightarrow p\)[/tex] ("If the area increases by a factor of 4, then the dimensions of the rectangle are doubled.") is true.
Thus, the two options that are true are:
- [tex]\(p \rightarrow q\)[/tex]
- [tex]\(q \rightarrow p\)[/tex]
1. Original Conditional ( [tex]\(p \rightarrow q\)[/tex] ):
- Statement: If [tex]\(p\)[/tex] represents doubling the dimensions of a rectangle and [tex]\(q\)[/tex] represents the area increasing by a factor of 4, then [tex]\(p \rightarrow q\)[/tex] is stated as "If the dimensions of a rectangle are doubled, then the area increases by a factor of 4."
- Truth Value: True
2. Inverse ( [tex]\(-p \rightarrow -q\)[/tex] ):
- Statement: If [tex]\(-p\)[/tex] represents not doubling the dimensions of a rectangle and [tex]\(-q\)[/tex] represents the area not increasing by a factor of 4, then [tex]\(-p \rightarrow -q\)[/tex] is stated as "If the dimensions of a rectangle are not doubled, then the area does not increase by a factor of 4."
- Truth Value: False
3. Converse ( [tex]\(q \rightarrow p\)[/tex] ):
- Statement: If [tex]\(q\)[/tex] represents the area increasing by a factor of 4 and [tex]\(p\)[/tex] represents doubling the dimensions of a rectangle, then [tex]\(q \rightarrow p\)[/tex] is stated as "If the area increases by a factor of 4, then the dimensions of the rectangle are doubled."
- Truth Value: True
4. Contrapositive ( [tex]\(-q \rightarrow -p\)[/tex] ):
- Statement: If [tex]\(-q\)[/tex] represents the area not increasing by a factor of 4 and [tex]\(-p\)[/tex] represents not doubling the dimensions of a rectangle, then [tex]\(-q \rightarrow -p\)[/tex] is stated as "If the area does not increase by a factor of 4, then the dimensions of the rectangle are not doubled."
- Truth Value: False
Based on these logical evaluations, the two true statements are:
1. [tex]\(p \rightarrow q\)[/tex] ("If the dimensions of a rectangle are doubled, then the area increases by a factor of 4.") is true.
2. [tex]\(q \rightarrow p\)[/tex] ("If the area increases by a factor of 4, then the dimensions of the rectangle are doubled.") is true.
Thus, the two options that are true are:
- [tex]\(p \rightarrow q\)[/tex]
- [tex]\(q \rightarrow p\)[/tex]
