To solve this problem, we need to examine each of the logical statements under the condition that the shape is a rectangle. Since a rectangle has four sides, the statement [tex]\( q \)[/tex] is true. Now, let's evaluate each of the logical expressions step-by-step given [tex]\( q \)[/tex] is true:
1. [tex]\( p \rightarrow q \)[/tex] (If a shape is a triangle, then it has four sides):
- [tex]\( p \)[/tex]: A shape is a triangle.
- [tex]\( q \)[/tex]: A shape has four sides.
- In a logical implication where [tex]\( p \rightarrow q \)[/tex], if [tex]\( p \)[/tex] is false (the shape is not a triangle) or [tex]\( q \)[/tex] is true (the shape has four sides), the implication [tex]\( p \rightarrow q \)[/tex] will be true. However, for completeness, let's evaluate it:
- In our context, if a shape is a triangle (which it isn't because it's a rectangle) then it should have four sides. This statement is not generally true because triangles do not have four sides. Hence, this statement is false.
2. [tex]\( p \wedge q \)[/tex] (A shape is a triangle and has four sides):
- [tex]\( p \)[/tex]: A shape is a triangle.
- [tex]\( q \)[/tex]: A shape has four sides.
- For this statement to be true, both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must be true simultaneously. But since a triangle cannot have four sides (it can only have three), this conjunction [tex]\( p \wedge q \)[/tex] is false.
3. [tex]\( p \leftrightarrow q \)[/tex] (A shape is a triangle if and only if it has four sides):
- [tex]\( p \)[/tex]: A shape is a triangle.
- [tex]\( q \)[/tex]: A shape has four sides.
- This biconditional statement means that [tex]\( p \)[/tex] and [tex]\( q \)[/tex] must either both be true or both be false.
- Here, [tex]\( p \)[/tex] (a shape being a triangle) and [tex]\( q \)[/tex] (a shape having four sides) cannot be both true or both false because they refer to mutually exclusive properties (triangles do not have four sides). Therefore, this biconditional statement is false.
4. [tex]\( q \rightarrow p \)[/tex] (If a shape has four sides, then it is a triangle):
- [tex]\( p \)[/tex]: A shape is a triangle.
- [tex]\( q \)[/tex]: A shape has four sides.
- This implication states that if a shape has four sides, it must be a triangle. However, shapes with four sides include rectangles, squares, parallelograms, and other quadrilaterals, not just triangles.
- Therefore, this implication is false.
Given our analysis, none of these logical statements hold true if the shape is a rectangle. Therefore, the correct conclusion is that none of the given statements are true. The answer to the question is:
[tex]\[ \boxed{0} \][/tex]
