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I'm doing some old exams and found following question:

Set $S={{1,2,3}}$ is given. Provide an example of binary operation in set S, binary relation in set S and a function $f:S\rightarrow R$.

So, I'm thinking about $+_4$ as operation, but wouldn't it be a relation too? I can take ordered pair from $S$ and for every ordered pair associate an element form $S$ with every pair.

For the third part, I'm thinking about $y=x*1.25$ or something similar, but that part isn't so problematic for me.

I did read the Wikipedia articles, but the difference between operations and relations isn't clear to me.

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up vote 4 down vote accepted

A binary operation is a function from $S\times S \to S$ such as addition, multiplication or anything really.

A binary relation is just a subset of $S^2$, that is not necessarily a function and it doesn't have to include all the elements of $S$ in one way or another.

A function $f\colon S\to R$ is a relation, this time it's a subset of $S\times R$ however it satisfies a certain property, if you take some $s \in S$ then there is only a unique ordered pair with $s$ in it, so if you have $\langle s,r_1\rangle$ as well $\langle s,r_2\rangle$ then you can say that $r_1 = r_2$.

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Thanks a lot! It is clear to me now. –  AndrejaKo Sep 4 '10 at 13:50
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