QQuestionMathematics
QuestionMathematics
What are the converse, inverse, and contrapositive of the following true conditional? What are the truth values of each?
If a statement is false, give a counterexample.
If a figure is a rectangle, then it is a parallelogram.
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Answer
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Step 1:I'll solve this step by step, carefully following the LaTeX formatting guidelines:
Step 2:: Identify the Original Conditional Statement
Original statement: $$p \implies q$$ (If a figure is a rectangle, then it is a parallelogram)
Let's define the original statement:
Step 3:: Determine the Converse
Final Answer
- Converse: q \implies p (If a figure is a parallelogram, then it is a rectangle) - Truth value: FALSE - Counterexample: A rhombus is a parallelogram but not a rectangle Step 3: Determine the Inverse - Inverse: \neg p \implies \neg q (If a figure is not a rectangle, then it is not a parallelogram) - Truth value: FALSE - Counterexample: A triangle is not a rectangle, but it's also not a parallelogram Step 4: Determine the Contrapositive The contrapositive negates both parts and reverses their order: - Contrapositive: \neg q \implies \neg p (If a figure is not a parallelogram, then it is not a rectangle) - Truth value: TRUE - Reasoning: This is logically equivalent to the original statement - Original statement: TRUE - Converse: FALSE - Inverse: FALSE - Contrapositive: TRUE
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