Answered

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Read the statement:

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 statements 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]\neg q \rightarrow \neg p[/tex] represents the converse of the original conditional statement.
E. [tex]\neg q \rightarrow p[/tex] represents the contrapositive of the original conditional statement.

Sagot :

GinnyAnswer
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]
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