To determine which logical statements are true if the shape is a rectangle, let us analyze the given propositions step-by-step.
Given:
- [tex]\( p \)[/tex]: A shape is a triangle.
- [tex]\( q \)[/tex]: A shape has four sides.
Let's consider each logical statement one by one:
### 1. [tex]\( p \rightarrow q \)[/tex]
This is the implication statement "If [tex]\( p \)[/tex] then [tex]\( q \)[/tex]".
- For a rectangle: [tex]\( p \)[/tex] (A shape is a triangle) is False.
- For a rectangle: [tex]\( q \)[/tex] (A shape has four sides) is True.
The implication [tex]\( p \rightarrow q \)[/tex] is True when [tex]\( p \)[/tex] is False, regardless of the value of [tex]\( q \)[/tex]. Hence, for a rectangle:
- [tex]\( p \rightarrow q \)[/tex] is True.
### 2. [tex]\( p \wedge q \)[/tex]
This is the conjunction statement "[tex]\( p \)[/tex] and [tex]\( q \)[/tex]".
- For a rectangle: [tex]\( p \)[/tex] (A shape is a triangle) is False.
- For a rectangle: [tex]\( q \)[/tex] (A shape has four sides) 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 for a rectangle, for a rectangle:
- [tex]\( p \wedge q \)[/tex] is False.
### 3. [tex]\( p \leftrightarrow q \)[/tex]
This is the biconditional statement " [tex]\( p \)[/tex] if and only if [tex]\( q \)[/tex]".
- For a rectangle: [tex]\( p \)[/tex] (A shape is a triangle) is False.
- For a rectangle: [tex]\( q \)[/tex] (A shape has four sides) is True.
The biconditional [tex]\( p \leftrightarrow q \)[/tex] is True if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] have the same truth value (both are True or both are False). Since [tex]\( p \)[/tex] is False and [tex]\( q \)[/tex] is True, for a rectangle:
- [tex]\( p \leftrightarrow q \)[/tex] is False.
### 4. [tex]\( q \rightarrow p \)[/tex]
This is the implication statement "If [tex]\( q \)[/tex] then [tex]\( p \)[/tex]".
- For a rectangle: [tex]\( q \)[/tex] (A shape has four sides) is True.
- For a rectangle: [tex]\( p \)[/tex] (A shape is a triangle) is False.
The implication [tex]\( q \rightarrow p \)[/tex] is True only if [tex]\( q \)[/tex] is False or [tex]\( p \)[/tex] is True. Since [tex]\( q \)[/tex] is True and [tex]\( p \)[/tex] is False, for a rectangle:
- [tex]\( q \rightarrow p \)[/tex] is False.
In summary, for a rectangle, the truth values of the logical statements are as follows:
- [tex]\( p \rightarrow q \)[/tex] : True
- [tex]\( p \wedge q \)[/tex] : False
- [tex]\( p \leftrightarrow q \)[/tex] : False
- [tex]\( q \rightarrow p \)[/tex] : False
