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.
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where are you stuck?
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.
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!
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But if your teacher doesn't require that you know them, its your choice.
what does ((c,d),(f,g)) tell you?
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
((a,b),(c,d)) has the property ad=bc ... not ab=cd
as I've said above, from ((c,d),(f,g)), we know that cg = df. Then..?
ad = bc
multiply by g
using cg=df we get
divide by d
you should look at the case where d=0 to validate that everything still works out.
Thank you. It is indeed trivial then. I'm sorry for the trouble, for some reason I just couldn't think of that earlier..
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.