Quotient by an equivalence relation

Alright kiddo, let's talk about something called "quotient by an equivalence relation."

Imagine you have a bunch of different toys, some are red, some are blue, and some are green. Now, let's say you want to group all the toys that are the same color together.

So, you put all the red toys in one pile, all the blue toys in another pile, and all the green toys in a third pile. Each pile is a group of toys that share the same color.

Now, let's say you have two toys that look a bit different from each other but they belong to the same group. For example, you have two red robots, one with round eyes and another with square eyes. Even though they look a bit different, they are still part of the group of red toys.

This is where the idea of equivalence comes in. The two robots may not look exactly the same but they are equivalent in color, just like how two circles may be different sizes but they are still equivalent in shape.

Quotient by an equivalence relation means taking a big group of things and grouping them into smaller groups based on their equivalence. In our example, we started with a big group of toys and divided them into smaller groups based on their color.

Now, we take each of these smaller groups and group them even further based on their equivalence. For example, we could take the red pile and group it further into two smaller piles, one for robots with round eyes and another for robots with square eyes.

This process of dividing into smaller groups based on equivalence is called "quotienting." We are creating smaller groups, or "quotients," based on the equivalence relation.

So, to sum it up: quotient by an equivalence relation is dividing a big group into smaller groups based on their equivalence, or how similar they are to each other. We keep doing this until we have smaller and smaller groups that are as similar as possible to each other.