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Math Help - converse & contrapositive

  1. #1
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    converse & contrapositive

    "Richmond will win the AFL premiership only if pigs can fly"

    Here is how ive gone about doing this:

    if p="pigs can fly" and r="Richmond will win the AFL premiership"
    then the above statement can be written as p-->r

    converse: (-p)-->(-r) If pigs can't fly, Richmond will not win the AFL premiership.

    contrapositive: (-r)-->(-p) If Richmond do not win the AFL premiership then pigs cannot fly


    Is what I have done correct? The answers given suggest otherwise, but I'm convinced that I haven't made a mistake. Can anyone clarify this?
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  2. #2
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    Quote Originally Posted by chrizzle View Post
    "Richmond will win the AFL premiership only if pigs can fly"

    Here is how ive gone about doing this:

    if p="pigs can fly" and r="Richmond will win the AFL premiership"
    then the above statement can be written as p-->r

    converse: (-p)-->(-r) If pigs can't fly, Richmond will not win the AFL premiership.

    contrapositive: (-r)-->(-p) If Richmond do not win the AFL premiership then pigs cannot fly


    Is what I have done correct? The answers given suggest otherwise, but I'm convinced that I haven't made a mistake. Can anyone clarify this?
    You have fine understanding of converse and contrapositive, you merely made a mistake with the "only if". The sentence "A only if B" is actually A-->B, not B-->A. Think about it, you should see why. On the other hand "A if B" is B-->A.
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  3. #3
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    I would like to add that AFAIK the converse of p -> r is r -> p, while ~p -> ~r is the contrapositive of the converse. Now, the contrapositive of any statement q is equivalent to q, so whether to consider r -> p or ~p -> ~r the converse of p -> r is a matter of nomenclature rather than logical equivalence.
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  4. #4
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    Quote Originally Posted by emakarov View Post
    I would like to add that AFAIK the converse of p -> r is r -> p, while ~p -> ~r is the contrapositive of the converse. Now, the contrapositive of any statement q is equivalent to q, so whether to consider r -> p or ~p -> ~r the converse of p -> r is a matter of nomenclature rather than logical equivalence.
    I've really been making a lot of silly errors lately. From what I understand ~p -> ~r is generally called the inverse of p -> r, even though as you say it is logically equivalent to the converse r -> p. Thanks for the correction.
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