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Math Help - Methods of Proof: Vacuous Proof

  1. #1
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    Methods of Proof: Vacuous Proof

    THIS IS NOT HOMEWORK! This is a practice problem from our textbook. Class link: Discrete Math I
    Methods of Proof: Vacuous Proof-untitled-1.jpg

    1) "By the definition of R we see that there are no ordered pairsin R where the first entry of one and the second entry of another is the
    same..."
    I do not understand what the definition of R is, nor what they mean by the first entry of one and the second entry of another is the same.

    2) "...so the hypothesis, ..., cannot be true."
    Maybe because I do not understand #1 above, I also do not understand this conclusion.

    3) "...this shows that the implication is true"
    I obviously do not understand this either.

    Any clarification would be greatly appreciated. Thank you!
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  2. #2
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    Re: Methods of Proof: Vacuous Proof

    Quote Originally Posted by sflink View Post
    1) "By the definition of R we see that there are no ordered pairsin R where the first entry of one and the second entry of another is the
    same..."
    I do not understand what the definition of R is, nor what they mean by the first entry of one and the second entry of another is the same.

    2) "...so the hypothesis, ..., cannot be true."
    Maybe because I do not understand #1 above, I also do not understand this conclusion.

    3) "...this shows that the implication is true"
    I obviously do not understand this either.
    For #1 there is nothing to understand. A relation is a set of ordered pairs. That is exactly what R is.

    For #2 are that two pair in R that look like (a,b),~(b,c).
    The answer is no. So (a,b)\in R\text{ and }(b,c)\in R is a false statement.

    For #3. The statement "It false then anything" is always true.
    So if (a,b)\in R\text{ and }(b,c)\in R then (a,c)\in R is TRUE.
    Because A false statement implies any statement.
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  3. #3
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    Re: Methods of Proof: Vacuous Proof

    Thank you!
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