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Math Help - Truth value of a quantifier statement

  1. #1
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    Question Truth value of a quantifier statement

    Hello,

    I need to determine whether the following statement is true of false:

    \forall x(x > 1\to x^2 > x) Domain: All reals


    I think the statement is true since I cannot find a value which would make the first part true while the second part false. However, I don't know how to prove the statement true without plugging in various numbers, but doing so would not prove that the statement is true in general, so I was wondering how one would start this question.

    Thanks
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  2. #2
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by Nostalgia View Post
    Hello,

    I need to determine whether the following statement is true of false:

    \forall x(x > 1\to x^2 > x) Domain: All reals


    I think the statement is true since I cannot find a value which would make the first part true while the second part false. However, I don't know how to prove the statement true without plugging in various numbers, but doing so would not prove that the statement is true in general, so I was wondering how one would start this question.

    Thanks
    Here are some thoughts

    Notice, x^{2}=xx>x \Leftrightarrow x>\frac{x}{x}=1.
    So, clearly a contradiction will arise if we assume x\leq\\1 for \forall\\x.
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  3. #3
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    Hello Nostalgia
    Quote Originally Posted by Nostalgia View Post
    Hello,

    I need to determine whether the following statement is true of false:

    \forall x(x > 1\to x^2 > x) Domain: All reals


    I think the statement is true since I cannot find a value which would make the first part true while the second part false. However, I don't know how to prove the statement true without plugging in various numbers, but doing so would not prove that the statement is true in general, so I was wondering how one would start this question.

    Thanks
    The proof is quite simple, provided we are given that we may multiply both sides of an inequality by a positive number; i.e. provided we know that:
    a > b and c > 0 \Rightarrow ac > bc
    For we simply multiply both sides by x, noting that x>1 \Rightarrow x >0. So:
    x>1 \Rightarrow xx > 1x \Rightarrow x^2>x
    Grandad
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