QQuestionMathematics
QuestionMathematics
The second statement is the $\qquad$ of the first.
\begin{aligned}
& x \Rightarrow y \\
& \neg y \Rightarrow \neg x
\end{aligned}
A. contradiction
B. converse
C. inverse
D. contrapositive
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Answer
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Step 1:I'll solve this step-by-step, focusing on the logical relationship between the two statements.
Step 2:: Understand the Given Statements
- First statement: $$x \Rightarrow y
- Second statement: \neg y \Rightarrow \neg x
Step 3:: Analyze the Logical Relationship
- This involves understanding the transformations of logical implication - The key is to identify how the original statement has been modified
Step 4:: Recall Logical Transformations
- Original implication: $$x \Rightarrow y
- Possible transformations include:
Step 5:
Converse: $$y \Rightarrow x
Step 6:
Inverse: $$\neg x \Rightarrow \neg y
Step 7:
Contrapositive: $$\neg y \Rightarrow \neg x
Step 8:: Identify the Correct Transformation
- The second statement is $$\neg y \Rightarrow \neg x
- This matches the definition of the contrapositive
Step 9:: Verify the Contrapositive Property
- This means $$x \Rightarrow y$$ is logically equivalent to $$\neg y \Rightarrow \neg x
- The contrapositive is logically equivalent to the original statement
Final Answer
Contrapositive
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