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Let X = {1,2,....,10} Define a relation R on X x X by (a,b)R(c,d) if a + d = b + c

I lose track of what it is asking on the part italicized. I have a similar question that ends in ad = bc as well with everything prior being similar.

Thanks for your help!

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I still do not understand, maybe someone could show me how it works with a different X, maybe X = { 23, 24, 25, ... 33} so that I can see how it works. I want to understand the concept fully, then I can apply it to the original X set. – Matt Apr 25 '13 at 3:37
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This relation is on pairs of integers. $(1,2)$ is related to $(3,4)$ because $1+4=2+3$. $(1,2)$ is not related to $(5,4)$ because $1+4\neq 2+5$. Hence, some pairs are related to other pairs. What you do with this information depends on what comes next.

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So I would have a set of all the pairs of elements that relate in that way? (5,1) (5,1) would also be in the set? – Matt Apr 25 '13 at 3:13
Do the elements of the set have to relate that way with all other elements of the set? or just one of the other elements in the set? – Matt Apr 25 '13 at 3:14
$(1,2)$ is related to $(1,2)$ because $1+2=2+1$. Also $(1,2)$ is related to $(10,11)$ because $1+11=2+10$. All such combinations... – vadim123 Apr 25 '13 at 3:18
{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8)(9,9)(10,10)} so (a.b)R(c,d) works for all these, but if I put (1,2) in the set it doesnt work with any of them... I feel like I am missing something here... sorry – Matt Apr 25 '13 at 3:26
You're right, any $(a,a)$ is related to any $(b,b)$, but $(1,2)$ isn't related to any $(a,a)$. You're not missing anything. It's not like if two elements aren't related you're doing something wrong. Some pairs are related to other pairs, and some aren't. – vadim123 Apr 25 '13 at 3:37

In words, what this is saying is that if you want to see if any two pairs of numbers in $X$ are related such as $(a,b)$ and $(c,d)$, for any $a,b,c,d \in X$, you find $a+d$ and see if it equates to $b+c$ if it does then the two are considered related by the definition of relation given.

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