Consider the following proposition: For all x, if Snazzy(x) then Nifty(x). Find the contrapositive of this proposition.
Understand the Problem
The question is asking for help with a logic problem regarding propositional logic and specifically finding the contrapositive of a given proposition. This involves understanding logical statements and their relationships.
Answer
The contrapositive is $\neg Q \rightarrow \neg P$.
Answer for screen readers
The contrapositive of the proposition "If P, then Q" is "If not Q, then not P", represented as $\neg Q \rightarrow \neg P$.
Steps to Solve
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Identify the Original Proposition First, we need to understand the original proposition. A typical proposition can be in the form "If P, then Q" which can be expressed as $P \rightarrow Q$.
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Define the Contrapositive The contrapositive of a statement "If P, then Q" is "If not Q, then not P". This can be expressed as $\neg Q \rightarrow \neg P$, where $\neg$ symbolizes "not".
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Convert the Original Proposition Using the structure of the original proposition, replace P and Q with the specific statements from the problem to express the contrapositive correctly.
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Write the Final Form The final contrapositive should be formatted clearly, showing the logical connection: $ \neg Q \rightarrow \neg P $.
The contrapositive of the proposition "If P, then Q" is "If not Q, then not P", represented as $\neg Q \rightarrow \neg P$.
More Information
Understanding the contrapositive is crucial in logic, as it is logically equivalent to the original statement, meaning both the statement and its contrapositive have the same truth value.
Tips
One common mistake is mixing up the contrapositive with the converse, which is "If Q, then P". Remember that the contrapositive negates both parts of the original proposition and reverses them.
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