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Math Help - Determine the truth values of the following statements

  1. #1
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    Determine the truth values of the following statements

    Let S(a,b) be the statement "ab=a/b".

    Given that the domain (or universe of discourse) for both variables consists of all positive, non-zero integers, what are the truth values of these two statements?

    a) ∀a∃b S(a,b)
    b) ∀b∃a S(a,b)

    Explain your reasoning when you answer.
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  2. #2
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    Quote Originally Posted by Runty View Post
    Let S(a,b) be the statement "ab=a/b".

    Given that the domain (or universe of discourse) for both variables consists of all positive, non-zero integers, what are the truth values of these two statements?

    a) ∀a∃b S(a,b)
    b) ∀b∃a S(a,b)

    Explain your reasoning when you answer.
    Well, we can put them into plain English:

    (a) For every integer a\geq 1 there is an integer b\geq 1 with ab=\frac{a}{b}.

    This is clearly true by choosing b=1 for any a.

    (b) For every integer b\geq 1, there is an integer a\geq 1 with ab=\frac{a}{b}.

    Now, regardless of a, it is clear that b^2=1. So if b=2 then (b) implies 4=1, which is a contradiction. So (b) is false.
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  3. #3
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    Quote Originally Posted by hatsoff View Post
    Well, we can put them into plain English:

    (a) For every integer a\geq 1 there is an integer b\geq 1 with ab=\frac{a}{b}.

    This is clearly true by choosing b=1 for any a.

    (b) For every integer b\geq 1, there is an integer a\geq 1 with ab=\frac{a}{b}.

    Now, regardless of a, it is clear that b^2=1. So if b=2 then (b) implies 4=1, which is a contradiction. So (b) is false.
    Thank you very much. Glad to get that one out of the way.
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