Monotone class

Imagine you have a big box full of toys. Some of the toys are red and some of the toys are blue. You want to sort the toys into two groups: one group of all the red toys and one group of all the blue toys.

Now, imagine you have a different box of toys. This time, the toys are sorted into three categories: cars, dolls, and stuffed animals. You want to sort the toys into two groups: one group of all the cars and dolls, and one group of all the stuffed animals.

In both of these scenarios, you are trying to group things together based on certain characteristics. This is similar to what mathematicians do when they work with a concept called a "monotone class".

A monotone class is basically a set of things that share some common property or characteristics. Mathematicians group together these things because they behave in similar ways when certain mathematical operations are applied to them.

For example, let's say you have a set of functions that are all "monotone increasing"; that is, they always increase in value as their input values increase. You could group these functions together in a "monotone class" because they share this property of always increasing.

Once you have a set of things grouped together in a monotone class, you can apply some rules or operations to them and get predictable outcomes. This is useful for solving all sorts of mathematical problems.

So, just like sorting your toys into colored groups or groups of different toy types, mathematicians use monotone classes to group things together so they can better understand how they behave under certain mathematical operations.