Let's analyze the truth values of each logical statement given the information that the shape is a rectangle.
1. Given: The shape is a rectangle.
- A rectangle has four sides.
- Therefore, [tex]\( q \)[/tex] is true (the shape has four sides).
- A rectangle is not a triangle.
- Therefore, [tex]\( p \)[/tex] is false (the shape is a triangle).
Now, let's evaluate each logical statement one by one:
1. [tex]\( p \rightarrow q \)[/tex] (If [tex]\( p \)[/tex], then [tex]\( q \)[/tex]):
- [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true.
- In logic, if the antecedent ([tex]\( p \)[/tex]) is false, the implication [tex]\( p \rightarrow q \)[/tex] is always true regardless of the truth value of [tex]\( q \)[/tex].
- Thus, [tex]\( p \rightarrow q \)[/tex] is true.
2. [tex]\( p \wedge q \)[/tex] ( [tex]\( p \)[/tex] and [tex]\( q \)[/tex]):
- [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true.
- The conjunction [tex]\( p \wedge q \)[/tex] is true only if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true.
- Since [tex]\( p \)[/tex] is false, [tex]\( p \wedge q \)[/tex] is false.
3. [tex]\( p \leftrightarrow q \)[/tex] ( [tex]\( p \)[/tex] if and only if [tex]\( q \)[/tex]):
- This means that [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must have the same truth value.
- [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true.
- Since [tex]\( p \)[/tex] and [tex]\( q \)[/tex] have different truth values, [tex]\( p \leftrightarrow q \)[/tex] is false.
4. [tex]\( q \rightarrow p \)[/tex] (If [tex]\( q \)[/tex], then [tex]\( p \)[/tex]):
- [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is false.
- The implication [tex]\( q \rightarrow p \)[/tex] is false when the antecedent ([tex]\( q \)[/tex]) is true and the consequent ([tex]\( p \)[/tex]) is false.
- Thus, [tex]\( q \rightarrow p \)[/tex] is false.
Summarizing the evaluations:
- [tex]\( p \rightarrow q \)[/tex]: true
- [tex]\( p \wedge q \)[/tex]: false
- [tex]\( p \leftrightarrow q \)[/tex]: false
- [tex]\( q \rightarrow p \)[/tex]: false
So, the truth values for the statements if the shape is a rectangle are:
[tex]\( (\text{True, False, False, False}). \)[/tex]
