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Math Help - T or F statements

  1. #1
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    T or F statements

    Find if statements are T or F and give explanation?

    Am I on the right tracK

    (\forall x \in R) (x >= -5)
    False. Not all real numbers are greater than or equal to -5. An example would be x=-4

    (\exists z \in R) (z < 1)
    True. There is, at least one, real number z that is less than one. An example: z = -1

    (\exists m \in R) (\forall x \in R) (xm = m)
    False. For example, let m=8 and x = -1. -8 != 8.

    (\exists t \in R) (\forall x \in R) (t < x )
    False. Let x = 1 and t=10
    Last edited by relyt; March 29th 2009 at 10:59 AM.
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  2. #2
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    Quote Originally Posted by relyt View Post
    Find if statements are T or F and give explanation?
    (\forall x \in R) (x >= 5)
    False. Not all real numbers are greater than or equal to -5. An example would be x=-4
    Where did the \color{red}-5 come from?


    (\exists t \in R) (\forall x \in R) (t < x )
    False. Let x = 1 and t=10
    You need a more general counter-example.
    Note that \color{red}(\forall t)[(t-1)<t]
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  3. #3
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    Sorry. That should be -5 and not 5 for the first statement
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  4. #4
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    Quote Originally Posted by relyt View Post
    Find if statements are T or F and give explanation?

    Am I on the right tracK

    (\forall x \in R) (x >= -5)
    False. Not all real numbers are greater than or equal to -5. An example would be x=-4

    (\exists z \in R) (z < 1)
    True. There is, at least one, real number z that is less than one. An example: z = -1

    (\exists m \in R) (\forall x \in R) (xm = m)
    False. For example, let m=8 and x = -1. -8 != 8.
    This says "there exist m". what if m= 0?

    (\exists t \in R) (\forall x \in R) (t < x )
    False. Let x = 1 and t=10
    As in the previous one (which is true) you can't just pick t to get a counter example. For all t, what happens if you take x= t?
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  5. #5
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    Ok, thanks.

    So the first one is false and the rest are true.

    For the "there exists" ones....as long as there is one number that makes the statement correct, then it is true?
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  6. #6
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    Quote Originally Posted by relyt View Post
    Ok, thanks.
    So the first one is false and the rest are true.

    For the "there exists" ones....as long as there is one number that makes the statement correct, then it is true?
    Absolutely NOT!
    The first is FALSE. x=-6<-5
    Second is TRUE. z=0<1
    Third is TRUE. m=0 works.
    The fourth is FALSE. (\forall t)[t-1<t] there is no smallest number.
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  7. #7
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    I think I'm a little confused now. Can we try one more and maybe someone can explain why this is true or false....and maybe what exactly I should look for first. Thanks and I appreciate the responses so far.

    (\forall x \in R) (\exists y \in R) (xy = 0.537)

    So this reads...for all real numbers x, there exists a real number y such that xy = 0.537
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  8. #8
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    Quote Originally Posted by relyt View Post
    I think I'm a little confused now. Can we try one more and maybe someone can explain why this is true or false....and maybe what exactly I should look for first. Thanks and I appreciate the responses so far.
    (\forall x \in R) (\exists y \in R) (xy = 0.537)
    So this reads...for all real numbers x, there exists a real number y such that xy = 0.537
    Zero is a real number. So what if x=0?
    What y would make the statement happen?
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  9. #9
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    There is no y that would...so it is false
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  10. #10
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    Quote Originally Posted by relyt View Post
    There is no y that would...so it is false
    Correct.
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