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In discrete math we define the relation as a sets of pairs of numbers, I understand when we write (a,b) we mean that (a) and (b) are realted what I don't grasp at all why the realtion between two diffrent is a subset of all ordered pairs between these sets. What is our intuition behind this definition?

Jalog_the_Martian
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  • It's just what you have said: Letting $R$ be the relation in question, $a\sim_R b$ is the same as saying that the ordered pair $(a,b)$ is in the relation $R$. – lulu Apr 22 '22 at 14:43
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    Do you mean "relation"? – kimchi lover Apr 22 '22 at 14:44
  • @lulu but we have said that (a) is realted to (b) but we have not define the realtion itself –  Apr 22 '22 at 14:44
  • @kimchilover yes –  Apr 22 '22 at 14:45
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    I don't understand. This is notation. It holds for any relation $R$. It does not define the relation. – lulu Apr 22 '22 at 14:45
  • Then you should correct your typo. – Paul Frost Apr 22 '22 at 14:46
  • @lulu so what really defines the relation because i don't understand the concept of realtion at all –  Apr 22 '22 at 14:47
  • Well, that's a different question. One which has nothing to do with the notation used to describe the relation. Presumably your reference materials have examples. And here is another discussion. – lulu Apr 22 '22 at 14:49
  • See also https://math.stackexchange.com/questions/518906/reflexivity-how-can-something-be-related-to-itself/518935#518935 – Michael Hoppe Apr 22 '22 at 18:56

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First, think about some relations that you will be very familiar with such as $=$ and $\le$.

Let us also for simplicity use our underlying set as the integers, $\mathbb{Z}$. If we say that our relation, $R$, is $=$ then we have that $nRm$ iff $n=m$. So the only ordered pairs in the corresponding subset of ordered pairs are those of the form $(n,n)$. So $R=\{(n,n) | n \in \mathbb{Z} \}$

Now, what about the relation $nRm$ iff $n\le m$.

As an exercise you tell me whether $(0,1), (8,3)$ or $(-2,-2)$ are in the corresponding subset.

Jalog_the_Martian
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  • it will be the ordered pair (0,1) –  Apr 22 '22 at 15:16
  • @Masoncarlos is that the only pair? – Jalog_the_Martian Apr 22 '22 at 15:17
  • i didn't notice the equal it will be (0,1) and (-2,-2) –  Apr 22 '22 at 15:18
  • Yeah indeed. So do you see how those ordered pairs are within the subset. So the relation is just the entire subset where the ordered pair has the appropriate form. – Jalog_the_Martian Apr 22 '22 at 15:24
  • but the ordered pairs (a,b) just says (a) and (b) is realted to each other how can we translate these ordered pairs to a realtion ? –  Apr 22 '22 at 15:38
  • i mean there is a diffrence between two things related to each other and the realtion itself –  Apr 22 '22 at 15:39
  • I think you are over stating this difference. The relation defines what things are related. The subset of ordered pairs is a list of all elements which are related and thus is precisely the relation itself – Jalog_the_Martian Apr 22 '22 at 17:08