Question

#8

Write a counterexample for the converse of the conditional statement you chose for the Venn Diagram in the previous question.

Answer 1

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.