1. ## A question about definition of "Binary Relation"

Hello everybody.

Is the following definition true?

A binary relation from a set A to a set B is a subset of Cartesian product of A and B, such that in all ordered pairs there is or there is not a certain connection between the first elements (that come from A)and the second elements(that come from B). And the names of the Relations (for example the relation "greater than") describes the connection between first and second elements of ordered pairs.

If it's a true definition, then introduce a source that contains it. And if is not true, describe why.

Thanks.

2. Originally Posted by Researcher
Hello everybody.

Is the following definition true?

A binary relation from a set A to a set B is a subset of Cartesian product of A and B, such that in all ordered pairs there is or there is not a certain connection between the first elements (that come from A)and the second elements(that come from B). And the names of the Relations (for example the relation "greater than") describes the connection between first and second elements of ordered pairs.

If it's a true definition, then introduce a source that contains it. And if is not true, describe why.

Thanks.
Hi.

An interesting question, and one I must admit I am not too sure about. I have certainly never seen a binary relation defined in such a way. However, I can see no reason as to why this is not valid. Essentially it says that if we take the set of ALL ordered pairs then some of them are in the subset, and others are not. That is to say, the sets are disjoint. I am pretty sure that that is all the definition is saying, and clearly it is true. (Note that I am using "clearly" in an entirely un-rigorous sense, but I can't think of any relation where this would fail).

For instance, take the binary relation $\leq$ on $\mathbb{Z} \times \mathbb{Z}$. Clearly either $(a,b) \in S$ or $(a,b) \notin S$ as $a \leq b$ or $a \nleq b$.

Also, the bit that you made blue is merely the author (or whoever gave you the definition) making life easier for themselves. It is not really part of the definition.

3. Thanks.
If we don't think of the blue part as a part of the definition, is it a true statement?

4. Hello Researcher
Originally Posted by Researcher
Thanks.
If we don't think of the blue part as a part of the definition, is it a true statement?
Yes. You'll find that this is a commonly used definition of a binary relation. For instance, see here.