Equivalence relation

An equivalence relation is like a way of putting things into groups where you treat some things similar to each other. Think of it like how you group your toys together - all the cars in one group, all the dolls in another group, and so on.

To say that two things are equivalent means they are the same in some way, or they have something in common. For example, you and your best friend may be different people, but you're equivalent in some ways because you both like playing together and have fun doing similar things.

For a relation to be an equivalence relation, it needs to satisfy three important rules. These are called reflexivity, symmetry, and transitivity.

Reflexivity means that each thing is always equivalent to itself. That's like saying each toy always belongs to its own group.

Symmetry means that if A is equivalent to B, then B is also equivalent to A. That's like saying if you think your friend is like you, then your friend should also think that you're like them.

Transitivity means that if A is equivalent to B and B is equivalent to C, then A is also equivalent to C. That's like saying if you think your friend is like you, and your friend thinks another kid is like them, then you should also think that the other kid is like you in some way.

So when you have a relation that follows all three of these rules, it's called an equivalence relation. And it's helpful because you can use it to compare things and put them into groups based on how similar they are to each other.