Sets equivalence relations

Question

I seem to be having a hard time understanding some basic sets concepts. In week 5 of my class, I learnt about the cross product of 2 sets to be the following $A \times B = \{(a,b) : a \in A, b \in B \} $

To give an example, this will look something like

$A = \{1,2\} \text{ B } = \{x, y,z \} \text{ then } A \times B = \{ (1, x), (1, y), (1, z), (2, x), (2, y), (2, z) \}$

Now fast forward to the middle of the semester where we are talking about equivalence relation and partitions. He made the case that for a set $S$, $R \subseteq S \times S$

I am having a bit a of hard time understanding why we have $S \times S$ if we are only given one set. Partly because I only see one set $S$ and not two sets.

Any help in breaking down the concepts would be greatly appreciated. Basically how do you explain something like this to your 10 year old.

Answer 1

The notion of "equivalence" comes to generalize "equality". If two objects are equal if and only if they are the same, equivalence relations come to relax that strict demand. For example, two people are equivalent if and only if they have the same height; or the same car; or the same biological mother.

These are all properties by which we can compare people and liken them to one another, even if they are not the same person.

Going back to the mathematical world, equivalence relation generalize the notion of equality, but for that we need a fixed domain of discourse. And we want to compare two elements from that domain of discourse. If "similar" means that $(x,y)\in R$ then we want both $x$ and $y$ to be from our domain. In particular this means that $R$ is a subset of $S\times S$.

And if what bothers you is that we take the product of $S$ with itself, and not with another set, remember that $2\times 2$ is a well-defined notion in the natural number. Cartesian product, much like multiplication, is defined for any two [suitable] objects, not necessary distinct two.

Answer 2

An equivalence relation is a rule which sets certain elements of the set $S$ "equal" to each other. Basically, it's just a way of partitioning the set into different pieces, where all the elements in each piece are equal to each other.

You can also write an equivalence relation as a collection of ordered pairs (a,b) for each pair of elements that satisfy aRb ("a related to b"). If you were to take the set R of all those ordered pairs, you would get something like {(a,b),(c,d),...}, which is a subset of the product $S\times S$. Remember that $S\times S$ is the set of all possible ordered pairs of elements of $S$, but only some of them will be in the set R making it a subset. Hope that helps.