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Math Help - Negation Question

  1. #1
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    Negation Question

    In my discrete math class I was given the following problem

    Let A and B be sets of real numbers. Write the negation of
    each of the following statements:
    (a) For every a ∈ A, it is true that a^2 ∈ B.
    (b) For at least one a∈ A,it is true that a^2 ∈ B.

    Would the negation for each question be correct if I said:

    For every a that is not an element of A it is not true that a^2 is not an element of B

    and

    For at least one a that is not an element of A it is not that that a^2 is not an element of B

    ANy help provided would be appreciated. Also does anyone know any good books for discrete math to use as a reference? The class I am in does not have a required textbook we just work off your instructors notes.

    Thanks
    and
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  2. #2
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    This is the negation of (a): \left( {\exists x \in A} \right)\left[ {x^2  \notin B} \right]
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  3. #3
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    Consider a statement \mathcal{Q}_1x_1\in A_1\dots\mathcal{Q}_nx_n\in A_n\,P(x_1,\dots,x_n) where each \mathcal{Q}_i is either \forall or \exists. Then its negation is \bar{\mathcal{Q}}_1x_1\in A_1\dots\bar{\mathcal{Q}}_nx_n\in A_n\,\neg P(x_1,\dots,x_n), where \bar{\exists} is \forall and \bar{\forall} is \exists.
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