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Math Help - Basic Question on Negation, Inverse and Contrapositive

  1. #1
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    Basic Question on Negation, Inverse and Contrapositive

    Hello to all.

    In a routine exercise I have been given the following statement:

    Let k be an integer, if k^2 is a multiple of 2, then k is a multiple of 2.

    I am then asked to determine the negative, converse, inverse and contrapositive of the aforementioned statement. The difficulty or doubt I am having is with the first portion of the statement (Let k be an integer...), specifically how or if it varies.

    The following are the rules I used to determine my answers:
    Starting with A=>B,

    Negation: not (A=>B) is log.equiv. to (A and not B)
    Converse: B=>A
    Inverse: (not A) => (not B)
    Contrapositive: (not B) => (not A)

    With that in mind, the following are my findings:

    Negation:
    Let k be an integer, if k^2 is a multiple of 2 and k is not a multiple of 2.

    Converse:
    Let k be an integer, if k is a multiple of 2, then k^2 is a multiple of 2.

    Inverse:
    Let k be an integer, if k^2 is not a multiple of 2 then k is not a multiple of 2.

    I wish to thank everyone in advance who offers help in determining if I am on the right path. Good day to all.
    Contrapositive:
    Let k be an integer, if k is not a multiple of 2 then k^2 is not a multiple of 2.
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  2. #2
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    Quote Originally Posted by gate13 View Post
    In a routine exercise I have been given the following statement:
    Let k be an integer, if k^2 is a multiple of 2, then k is a multiple of 2.
    I am then asked to determine the negative, converse, inverse and contrapositive of the aforementioned statement.
    Negation:
    Let k be an integer, if k^2 is a multiple of 2 and k is not a multiple of 2.
    You have one mistake. There is no if in the negation.

    The negation: There is an integer J such that J^2 is a multiple of 2 and J is not a multiple of 2.
    Last edited by Plato; January 9th 2010 at 01:38 PM.
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  3. #3
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    Thank you Plato for your quick response. Have a good day.
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