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Math Help - Negation clarification

  1. #1
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    Question Negation clarification

    x is a positive odd integer --> x + 2 is a positive odd integer


    The trouble comes in negating each statement when finding the contrapositive

    do i negate the statements to:

    x is a NEGATIVE odd integer
    x + 2 is a NEGATIVE odd integer

    or

    x is a positive EVEN integer
    x + 2 is a positive EVEN integer

    or

    x is a NEGATIVE EVEN integer
    x + 2 is a NEGATIVE EVEN integer

    or (likely not but i just thought i'd ask) U = universe
    x is a positive odd U-Z
    x + 2 is a positive odd U-Z

    I know that a statement has a truth value, but my prof and the text never went over what parts of a statement are eligible for negation. Is it right to say that the statements I have used as examples are in fact statements even though x is unknown, because they are quantified?

    Am I correct to say that the quantification of these statements is implicit rather than explicit?

    Argh this seems like a lot of questions, I'd really appreciate it someone could answer at least a few of them if answering them all is too laborious.

    Thanks
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Noxide View Post
    x is a positive odd integer --> x + 2 is a positive odd integer


    The trouble comes in negating each statement when finding the contrapositive
    Negating a statement has nothing to do with positive or negative numbers. If A \Rightarrow B then the contrapositive is \text{not }B \Rightarrow \text{ not } A.

    So here, "if x is a positive odd integer then x + 2 is a positive odd integer" is negated by putting "not" before the condition in each statement and switching them. That is, "if x+2 is not a positive odd integer then x is not a positive odd integer".
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  3. #3
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    Thanks.

    Can you tell me a little bit more about the condition of a statement. ie How it can be found more easily.

    Idk why my text doesn't go over this.
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