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Math Help - Discrete Math problem

  1. #1
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    Discrete Math problem

    dd
    Last edited by chillerbros17; February 28th 2007 at 03:17 PM.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by chillerbros17 View Post
    4. Let the domain D be the set of nonzero real numbers. D is the domain for both x and y. For each part, decide whether the statement is true or false. If statement is true, explain why. If statement is false, find negation and explain why negation is true.

    a. (There exists x) (for all y) xy= 1
    b. (For all x) (There exists y) xy=1
    c. (For all x) (for all y) [ sin^2x + cos^2x = sin^2y + cos^2y ]
    d. (There exists x) (There exists y) [ 2x+y = 5 ^ x-3y = -8 ]
    a) False. Let y = 0. Then x does not exist.
    b) False. Same argument as in a).
    c) True. [sin(x)]^2 + [cos(x)]^2 = 1 for all x (and similarly for all y).
    d) True. See attached graph.
    Attached Thumbnails Attached Thumbnails Discrete Math problem-intersection.jpg  
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    Quote Originally Posted by topsquark View Post
    a) False. Let y = 0. Then x does not exist.
    b) False. Same argument as in a).
    Since x and y are dreawn from the non-zero reals both of these are in fact true.

    RonL
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    Quote Originally Posted by topsquark View Post
    d) True. See attached graph.
    False as the curves do not meet when =-8.

    RonL
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Since x and y are dreawn from the non-zero reals both of these are in fact true.

    RonL
    (explicative) I hate it when I miss details like that.

    -Dan
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    False as the curves do not meet when =-8.

    RonL
    2x+y = 5 ^ x-3y = -8

    2x + y = -8
    y = -2x - 8

    5^x - 3y = -8
    3y = 5^x + 8

    y = (1/3)*5^x + 8/3

    which are the two functions I graphed.

    -Dan
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  7. #7
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    Quote Originally Posted by chillerbros17 View Post
    4. Let the domain D be the set of nonzero real numbers. D is the domain for both x and y. For each part, decide whether the statement is true or false. If statement is true, explain why. If statement is false, find negation and explain why negation is true.
    a. (There exists x) (for all y) xy= 1
    b. (For all x) (There exists y) xy=1
    Proposition (a) is false. It says that there is a number in the domain that has the property that it times any other number in the domain equals 1: clearly false.
    Its negation is (For all x)(there exists a y)[xy<>1].

    Proposition (b) is of course true for the nonzero reals are an Abelian group.
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  8. #8
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    What is the negation of (There exists x) (There exists y) [ 2x+y = 5 ^ x-3y = -8 ]
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  9. #9
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    Quote Originally Posted by chillerbros17 View Post
    What is the negation of (There exists x) (There exists y) [ 2x+y = 5 ^ x-3y = -8 ]
    [For all x][for all y]{2x+y<>5 OR x-3y<>-8} where "<>" means not equal.
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