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    Contrapositives of quantified subset statements

    E-MUN-1-6

    Area
    Topology
    Topics
    Fundamental Concepts
    Status
    draft

    Write the contrapositive of each of the statements of Exercise 5.

    Solution

    The contrapositive of an implication \(P\Rightarrow Q\) is \(\neg Q\Rightarrow\neg P\). Applying this to the statements of E-MUN-1-5 gives:

    1. \[ \bigl(x\notin A\text{ for every }A\in\mathcal A\bigr) \Longrightarrow x\notin\bigcup_{A\in\mathcal A}A. \]

    2. \[ \bigl(x\notin A\text{ for at least one }A\in\mathcal A\bigr) \Longrightarrow x\notin\bigcup_{A\in\mathcal A}A. \]

    3. \[ \bigl(x\notin A\text{ for every }A\in\mathcal A\bigr) \Longrightarrow x\notin\bigcap_{A\in\mathcal A}A. \]

    4. \[ \bigl(x\notin A\text{ for at least one }A\in\mathcal A\bigr) \Longrightarrow x\notin\bigcap_{A\in\mathcal A}A. \] A statement and its contrapositive have the same truth value, so by E-MUN-1-5, (a), (c), and (d) are true and (b) is false.

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