Defining relations $R$ and $S$ on $A \times B$?

Question

I've been trying to figure out a way to do this problem:

Let $A = \{-1, 1, 2, 4\}$ and $B = \{1, 2\}$ and define relations $R$ and $S$ on $A \times B$ as $$R = \{(-1, 1)(1,2)(1,2)(2,4)(4,1)(4,4)(2,2)(4,2)\}$$ $$S = \{(1,2),(2,2)\}$$

I found the set $A \times B$, and I know it relates to $R$ and $S$. I just don't understand what they mean by defining a relation. Do they want us to come up with a relation where we get the sets that are in $R$ and $S$? If so, how do we come up with a relation for $R$ and $S$?

Answer 1

A relation on $A$ and $B$ is a subset of $A \times B$.

Each element of $S$ and $R$ pairs, or relates, an element from $A$ and an element from $B$. There is little you can say about them since this concept is very general.

Answer 2

From the sound of it, the problem isn't asking you to define relations, it's giving you the definitions.

You might have seen this definition of an equivalence relation, but you can also think of an equivalence relation just as the collection of all related pairs.

Imagine your set $A$ is people, and your set $B$ is physical objects, and our relation says that $a \in A$ is related to $b \in B$ whenever person $a$ owns object $b$. You could think about this abstractly, like in the wikipedia article, or you could be a little more brute-force about it and decide to just list out all the relations.

$$\{ (\textrm{Tommy}, \textrm{truck}), (\textrm{Alice}, \textrm{arugula}), \cdots \}$$

This list defines the equivalence relation just as much as my above description. It isn't nearly as satisfying or as succinct, but you can use either the rule or the list of pairs to figure out what things are related.

In your problem, they're giving the definition of the equivalence relations $R$ and $S$ just by listing out all pairs of related objects. I assume the problem goes on to ask you to do something with those relations