What is a set, and how does it identify its members?

"A set is a selection of certain things out of a (normally larger) group. When we talk about a set, we’re declaring that certain specific items from that group are in the set, and certain items are not in the set. There’s no shades of gray: every element is either in or out. For instance, maybe the overall group I’m considering is my family, which consists of five people: Dad, Mom, Lizzy, T.J., and Marina. We could define one set — call it A — that contains Dad and Lizzy, but not the other three. Another set B might have Lizzy, T.J., and Marina in it, but not the two parents. The set C might have Dad and only Dad in it. The set D might have all five Davieses, and the set E might have nobody at all. Etc. You can see that every set is just a way of specifying which elements are in and which are out. Normally a set will be based on some property of its members, rather than just being some random assortment of elements. That’s what makes it worth thinking about. For example, the set P (for “parents”) might be “all the Davieses who are parents”: this set would contain Dad and Mom, and no one else. The set F (for “female”) might be declared as the female members, and contain Mom, Lizzy and Marina. The set H (for “humans”) would contain all five elements of the group. And so on."