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Contrapositive Statement

  1. Mar 14, 2009 #1
    The statement is:
    If α is one-to-one and β is onto, then βoα is one-to-one and onto.
    One-to-one is injection, onto is surjection, and being both is bijection. After showing that the statement is false, the contrapositive was asked for. The answer given is:
    If βoα is not one-to-one and onto, then α is not one-to-one or β is not onto.
    They changed the "and" to an "or." I was thinking that the "and" would be conserved in the contrapositive statement. Is it valid or necessary to change an "and" to an "or" for contrapositive statements?
     
  2. jcsd
  3. Mar 14, 2009 #2
    The opposite of "A and B" is "not A or not B". This is one of http://en.wikipedia.org/wiki/DeMorgan%27s_laws" [Broken]. So yes, you need to change "and" to "or" in this case when forming the contrapositive.
     
    Last edited by a moderator: May 4, 2017
  4. Mar 14, 2009 #3

    HallsofIvy

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    The contrapositive of "if A then B" is "if not B then not A",

    But your question is not really about the contrapositive, it is about "Not (A and B)".

    "Not (A and B)" is the same as saying "(not A or (not B)".

    This is because "A and B" is true only if A= T and B= T. If A= T, B= F; A= F, B= T; or A= F, B= F, "A and B" is false. "Not (aA and B)" must be true in exactly those cases. In particular, it must be true in the cases A= T, B= F and A= F, B= T. That is precisely "(not A) or (not B)".

    "(Not A) and (Not B)" would be true only in the case A= F, B= F.
     
  5. Mar 14, 2009 #4

    matt grime

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    No, the contrapositive is as stated: it is an or. You are negating things. The negation of

    A and B

    is

    not A or not B

    so it is both necessary and valid.

    Think about it: suppose A and B together imply C

    Then "not C" can only happen if at least one of A or B is not true, and that's not A or not B.
     
  6. Mar 14, 2009 #5
    So does that imply the above quote would equivalently be written as If βoα is not one-to-one or βoα is not onto...?
     
  7. Mar 15, 2009 #6
    Correct.
     
  8. Mar 15, 2009 #7
    Thanks
     
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