Answer :
Let's analyze each statement step by step given the fact that the shape is a rectangle:
1. [tex]\( p \rightarrow q \)[/tex] (If [tex]\( p \)[/tex], then [tex]\( q \)[/tex])
- [tex]\( p \)[/tex]: A shape is a triangle.
- [tex]\( q \)[/tex]: A shape has four sides.
- For a rectangle, [tex]\( p \)[/tex] (the shape being a triangle) is false.
- [tex]\( q \)[/tex] (the shape having four sides) is true.
The implication [tex]\( p \rightarrow q \)[/tex] (if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]) is true when:
- [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true.
- [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is false.
- [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is true.
- The only case it is false is if [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false.
Since [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true, [tex]\( p \rightarrow q \)[/tex] is true.
2. [tex]\( p \wedge q \)[/tex] ( [tex]\( p \)[/tex] and [tex]\( q \)[/tex])
- This statement is true if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true.
- Since [tex]\( p \)[/tex] (the shape being a triangle) is false for a rectangle and [tex]\( q \)[/tex] (the shape having four sides) is true, the conjunction [tex]\( p \wedge q \)[/tex] (false and true) is false.
3. [tex]\( p \leftrightarrow q \)[/tex] ([tex]\( p \)[/tex] if and only if [tex]\( q \)[/tex])
- This statement is true if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are either true or both are false.
- Since [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true, [tex]\( p \leftrightarrow q \)[/tex] (false if and only if true) is false.
4. [tex]\( q \rightarrow p \)[/tex] (If [tex]\( q \)[/tex], then [tex]\( p \)[/tex])
- [tex]\( q \)[/tex]: A shape has four sides (true for a rectangle).
- [tex]\( p \)[/tex]: A shape is a triangle (false for a rectangle).
The implication [tex]\( q \rightarrow p \)[/tex] (if [tex]\( q \)[/tex], then [tex]\( p \)[/tex]) is true when:
- [tex]\( q \)[/tex] is false and [tex]\( p \)[/tex] is false.
- [tex]\( q \)[/tex] is false and [tex]\( p \)[/tex] is true.
- [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is true.
- The only case it is false is if [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is false.
Since [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is false, [tex]\( q \rightarrow p \)[/tex] is false.
Based on the analysis, the statements evaluated as follows are:
- [tex]\( p \rightarrow q \)[/tex]: True
- [tex]\( p \wedge q \)[/tex]: False
- [tex]\( p \leftrightarrow q \)[/tex]: False
- [tex]\( q \rightarrow p \)[/tex]: False
Thus, the statement that is true if the shape is a rectangle is [tex]\( p \rightarrow q \)[/tex].
1. [tex]\( p \rightarrow q \)[/tex] (If [tex]\( p \)[/tex], then [tex]\( q \)[/tex])
- [tex]\( p \)[/tex]: A shape is a triangle.
- [tex]\( q \)[/tex]: A shape has four sides.
- For a rectangle, [tex]\( p \)[/tex] (the shape being a triangle) is false.
- [tex]\( q \)[/tex] (the shape having four sides) is true.
The implication [tex]\( p \rightarrow q \)[/tex] (if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]) is true when:
- [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true.
- [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is false.
- [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is true.
- The only case it is false is if [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false.
Since [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true, [tex]\( p \rightarrow q \)[/tex] is true.
2. [tex]\( p \wedge q \)[/tex] ( [tex]\( p \)[/tex] and [tex]\( q \)[/tex])
- This statement is true if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true.
- Since [tex]\( p \)[/tex] (the shape being a triangle) is false for a rectangle and [tex]\( q \)[/tex] (the shape having four sides) is true, the conjunction [tex]\( p \wedge q \)[/tex] (false and true) is false.
3. [tex]\( p \leftrightarrow q \)[/tex] ([tex]\( p \)[/tex] if and only if [tex]\( q \)[/tex])
- This statement is true if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are either true or both are false.
- Since [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true, [tex]\( p \leftrightarrow q \)[/tex] (false if and only if true) is false.
4. [tex]\( q \rightarrow p \)[/tex] (If [tex]\( q \)[/tex], then [tex]\( p \)[/tex])
- [tex]\( q \)[/tex]: A shape has four sides (true for a rectangle).
- [tex]\( p \)[/tex]: A shape is a triangle (false for a rectangle).
The implication [tex]\( q \rightarrow p \)[/tex] (if [tex]\( q \)[/tex], then [tex]\( p \)[/tex]) is true when:
- [tex]\( q \)[/tex] is false and [tex]\( p \)[/tex] is false.
- [tex]\( q \)[/tex] is false and [tex]\( p \)[/tex] is true.
- [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is true.
- The only case it is false is if [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is false.
Since [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is false, [tex]\( q \rightarrow p \)[/tex] is false.
Based on the analysis, the statements evaluated as follows are:
- [tex]\( p \rightarrow q \)[/tex]: True
- [tex]\( p \wedge q \)[/tex]: False
- [tex]\( p \leftrightarrow q \)[/tex]: False
- [tex]\( q \rightarrow p \)[/tex]: False
Thus, the statement that is true if the shape is a rectangle is [tex]\( p \rightarrow q \)[/tex].
