Consider the statement, “All quadrilaterals have four sides.” Its inverse is
The Correct Answer and Explanation is:
Correct Answer:
The inverse of the statement “All quadrilaterals have four sides” is:
“If a figure is not a quadrilateral, then it does not have four sides.”
Explanation
To understand how to form the inverse of a logical statement, we need to examine its structure closely. The given statement is:
“All quadrilaterals have four sides.”
This is a universal affirmative statement, and it can be rewritten in conditional (if-then) form for clarity:
“If a figure is a quadrilateral, then it has four sides.”
This type of sentence follows a standard logical structure:
- Original (conditional): If P, then Q
(P = “a figure is a quadrilateral”, Q = “it has four sides”)
The inverse of a conditional statement is formed by negating both the hypothesis (P) and the conclusion (Q) of the original statement:
- Inverse: If not P, then not Q
= “If a figure is not a quadrilateral, then it does not have four sides.”
This is exactly what we found earlier.
It is important to note that the truth of a statement does not automatically imply the truth of its inverse. Just because the original is true (all quadrilaterals have four sides), the inverse is not necessarily true. There are non-quadrilateral figures that also have four sides — for example, a four-sided figure that is not closed might not be considered a quadrilateral, depending on context. However, figures like trapezoids and rectangles, which are quadrilaterals, certainly have four sides.
To summarize:
- The inverse is made by negating both the “if” and “then” parts of the original statement.
- The inverse of “If a figure is a quadrilateral, then it has four sides” is:
“If a figure is not a quadrilateral, then it does not have four sides.”
