To provide a counterexample for the converse of a given conditional statement, we first need to define that statement. Let's say our original conditional statement is:
Original Conditional Statement: If \( A \) then \( B \) (Symbolically, \( A \rightarrow B \))
The converse of this statement is:
Converse Statement: If \( B \) then \( A \) (Symbolically, \( B \rightarrow A \))
A counterexample to the converse is a situation where \( B \) is true, but \( A \) is false.
Example
- Original Statement: If a shape is a square, then it is a rectangle. (\( A \) = "is a square", \( B \) = "is a rectangle")
- Converse Statement: If a shape is a rectangle, then it is a square. (\( B \rightarrow A \))
Counterexample
Consider a rectangle that is not a square, such as one with different lengths for its sides (e.g., length = 5 and width = 3). In this case:
- The shape is a rectangle (true for \( B \))
- The shape is not a square (false for \( A \))
Thus, the counterexample shows that the converse statement is not universally true, as there exist rectangles that are not squares. Therefore, the existence of rectangles (not squares) disproves the converse statement.
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