Questions: Consider the following statement. Every parallelogram is a quadrilateral. (a) Write the converse of the given statement. (b) Write the inverse of the given statement. (c) Write the contrapositive of the given statement.

Solution Steps

To solve this problem, we need to understand the definitions of converse, inverse, and contrapositive statements in logic.

1. Converse: The converse of a statement "If P, then Q" is "If Q, then P".
2. Inverse: The inverse of a statement "If P, then Q" is "If not P, then not Q".
3. Contrapositive: The contrapositive of a statement "If P, then Q" is "If not Q, then not P".

Given the statement "Every parallelogram is a quadrilateral" (If a figure is a parallelogram, then it is a quadrilateral):

(a) The converse would be "If a figure is a quadrilateral, then it is a parallelogram". (b) The inverse would be "If a figure is not a parallelogram, then it is not a quadrilateral". (c) The contrapositive would be "If a figure is not a quadrilateral, then it is not a parallelogram".

The original statement is "Every parallelogram is a quadrilateral," which can be expressed in logical form as: \[ P \implies Q \] where \( P \) is "a figure is a parallelogram" and \( Q \) is "a figure is a quadrilateral."

The converse of the statement is formed by reversing the implication: \[ Q \implies P \] Thus, the converse is: "If a figure is a quadrilateral, then it is a parallelogram."

The inverse of the statement negates both the hypothesis and the conclusion: \[ \neg P \implies \neg Q \] Therefore, the inverse is: "If a figure is not a parallelogram, then it is not a quadrilateral."

The contrapositive negates and reverses the original statement: \[ \neg Q \implies \neg P \] Consequently, the contrapositive is: "If a figure is not a quadrilateral, then it is not a parallelogram."

Final Answer