Subadditive set function

Okay, imagine you have a lot of toys and other cool things to play with. Let's say that you love playing with toys so much that you can't just play with one toy for a long time, you get bored quickly.

Now, let's say you have a mom or dad that tells you that you can only play with a certain number of toys each day. Maybe they say you can only choose two toys out of all the ones you have to play with for the day.

A subadditive set function is like your mom or dad setting these rules. It's a fancy way of saying that if you have a bunch of things that you can choose from, you can't play with all of them at once. Instead, you have to choose only some of them to play with.

Here's another example: Imagine there's a toy store with all kinds of toys in it. The store owner says that you can only pick out a certain number of toys, let's say five, and each toy has a certain value.

If you pick out one toy, it might be worth 5 points. If you pick out two toys, they might be worth 8 points together. But if you try to pick out three toys, they might only be worth 10 points, instead of 12 like you might think.

This is because subadditive set functions have a property that says that if you take any two sets of toys and add them together, you'll always get a set that's worth either the same amount or less than the sum of the two original sets.

So even though it might seem like picking more toys would always mean you're getting more points, this isn't necessarily true with subadditive set functions. It's like your mom or dad making sure you're not getting too carried away with your playtime, or the toy store owner making sure you're not taking too many valuable toys.