What are multisets under various equivalence relations?

A multiset is like a bag where you can have more than one of the same thing, kind of like your toy box when you have five red balls and three blue ones.

Imagine you're sorting your crayons. A regular set would be like having just one of each color, but a multiset is like having multiple crayons of the same color. You could have two green crayons, four yellow ones, and so on, it’s all about how many times something appears.

Now, imagine you're playing a game where you need to group your crayons based on some rules. These rules are called equivalence relations.

• If the rule is that "all red things are the same," then even if you have five red crayons, they all count as one group.
• If the rule is "each crayon counts individually," then every single crayon matters, even the ones with the same color.

So, depending on the rule you use to sort your crayons (or anything else), a multiset can look different. That’s what it means to understand multisets under various equivalence relations, it's like playing with different sorting games and seeing how your toy box changes!