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Show that the operation $[a,b]∙[c,d]=[ac+bd,ad+bc]$ is well defined in the construction of the set of integers.

$[a,b]$ is the representation of an integer as an ordered pair of whole numbers

Note: do not depend on the choice of the representatives of the ~−classes

Workings:

Let $[a,b]$ ~ $[a',b']$ and $[c,d]$ ~ $[c',d']$ so that

$[a',b'] [c',d'] = [a'c'+b'd,a'd'+b'c']$

Need to prove taht

$[a'c'+b'd,a'd'+b'c'] = [ac+bd,ad+bc]$

Now I am not sure what to do

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    You should say, let $[a,b]$ and $[a',b']$ represent the same integer, and $[c,d]$ and $[c',d']$ represent the same integer. Then you need to prove the equality you wrote. – 2017-01-19
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    My ~ didn't show up was the problem. THanks. – 2017-01-19
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    okay, so what does it mean for $[a,b]$ and $[a',b']$ to represent the same integer? – 2017-01-19
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    That though $a$ and $a'$ and $b$, $b'$ are different when taking $a,b$ or $a',b'$ together they represent the same integer. – 2017-01-19
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    well, right, but what does it mean to represent the same integer? What is the equivalence relation? – 2017-01-19

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