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windsylph

Prove that if an ordered pair of ordered pairs ((a,b),(c,d)) has the property ad=bc, then this property is transitive, i.e. if another ordered pair((c,d),(f,g)) is in this relation, then ((a,b),(f,g)) is in this relation as well, such that ag = bf.

  • one year ago
  • one year ago

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  1. Zarkon
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    where are you stuck?

    • one year ago
  2. windsylph
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    The actual proof, I know I have to show that ag = bf. But how do you actually show that knowing ad = bc and cg = df? Haha I know too that this might be trivial, but for some reason I just can't wrap my mind around it.

    • one year ago
  3. micrad25
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    Proofs can be difficult because there are so many to choose from but what i did was i made flashcards on all the proofs (in the back of the book) and memorized them. There is a lot but if you memorize one a day you will be a pro by the end of the year!

    • one year ago
  4. micrad25
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    But if your teacher doesn't require that you know them, its your choice.

    • one year ago
  5. Zarkon
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    what does ((c,d),(f,g)) tell you?

    • one year ago
  6. anonymoustwo44
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    let (a,b), (c,d), and (f,g) be any ordered pairs, such that ((a,b),(c,d)) and ((c,d),(f,g)) so that ab=cd and cd=fg. Now we know from basic algebra, ab=fg, hence ((a,b),(f,g)) when ((a,b),(c,d)) and ((c,d),(f,g)), therefore the property is transitive

    • one year ago
  7. Zarkon
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    @anonymoustwo44 ((a,b),(c,d)) has the property ad=bc ... not ab=cd

    • one year ago
  8. windsylph
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    as I've said above, from ((c,d),(f,g)), we know that cg = df. Then..?

    • one year ago
  9. Zarkon
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    ad = bc multiply by g adg=bcg using cg=df we get bcg=bdf thus adg=bdf divide by d ag=bf you should look at the case where d=0 to validate that everything still works out.

    • one year ago
  10. windsylph
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    Thank you. It is indeed trivial then. I'm sorry for the trouble, for some reason I just couldn't think of that earlier..

    • one year ago
  11. windsylph
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    Also, I forgot to remark that the components of the ordered pairs should only be positive integers, so I don't have to show the case where d = 0.

    • one year ago
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