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Consider these statements: p: The weather is not cloudy; q: It is not going to rain. If p => q, which of the following is a valid conclusion?

Mathematics

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Step-by-step solution

3 steps
  1. Identify the given statements and the implication.

    The given statements are:
    pp: The weather is not cloudy.
    qq: It is not going to rain.
    The implication is p⇒qp \Rightarrow q, which means "If the weather is not cloudy, then it is not going to rain."

  2. Determine the negations of the statements.

    The negation of pp is ¬p\neg p: The weather is cloudy.
    The negation of qq is ¬q\neg q: It is going to rain.

  3. Apply the rule of contrapositive.

    A valid conclusion from an implication p⇒qp \Rightarrow q is its contrapositive, which is ¬q⇒¬p\neg q \Rightarrow \neg p.
    Substituting the statements:
    ¬q⇒¬p\neg q \Rightarrow \neg p means "If it is going to rain, then the weather is cloudy."

    A valid conclusion is:

    If it is going to rain, then the weather is cloudy.\boxed{\text{If it is going to rain, then the weather is cloudy.}}
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