Homework Help: Definition of a relation

1. Jan 15, 2009

1. The problem statement, all variables and given/known data
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 $$s \in dom(R)$$.

2. Relevant equations
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3. 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?

2. Jan 15, 2009

tiny-tim

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 …

3. Jan 15, 2009

HallsofIvy

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.