What is a relation and how is it defined on a set?

[icantadd]

Homework Statement

This is a seemingly subtle point here, that would actually clear up both of the two previous posts I have made. A relation R is said to be defined on S and T if $$s \in S$$ and $$s \in dom(R)$$.

Homework Equations

na

The Attempt at a Solution

Does this mean, that if I see a question that starts if R is defined on S ... that I can assume if I define a relation on S, call it T, that the domain of T must also be S. Or for any relation that we define on a set, it can be assumed that the domain of the relation is that set?

 

[tiny-tim]

Hi icantadd!

Sorry, I'm not following any of that.

A relation on S is a subset of S x S.

From the PF Library page on relation …

A relation on a set A is a subset R of $$A \times A$$.
For a relation $$R \subseteq A\times A,~\text{and}~x, y \in A$$, we say xRy, i.e. x is related to y, if $$\left(x,y\right ) \in \mathbb{R}$$

 

[HallsofIvy]

Homework Statement

This is a seemingly subtle point here, that would actually clear up both of the two previous posts I have made. A relation R is said to be defined on S and T if $$s \in S$$ and $$s \in dom(R)$$.

Homework Equations

na

The Attempt at a Solution

Does this mean, that if I see a question that starts if R is defined on S ... that I can assume if I define a relation on S, call it T, that the domain of T must also be S. Or for any relation that we define on a set, it can be assumed that the domain of the relation is that set?

A relation on S is any subset of the cartesian product SxS, the set of ordered pairs of objects from S. It does not follow from that that every member of S must be in some ordered pair. For example, I could define R on Z, the set of integers by "xRy is x and y are both odd numbers" That would consist of things like (1, 1), (3, 5), (-3, 7), etc. That is also of course, a relation on "O", the set of odd integers. If R is a relation on both sets S and T, the members of the pairs of R must be contained in both S and T: some subset of the intersection of S and T.