2.1: The idea of a set

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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 Johnny. 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 Johnny 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 and Lizzy. The set \(H\) (for “humans") would contain all five elements of the group. And so on.

As with most of math, it turns out to be useful to define symbols for these concepts, because then we can talk about them more precisely and concisely. We normally list the members of a set using curly braces, like this: \[A = \{~\text{Dad}, \text{Lizzy}~\}\] or \[B = \{~\text{Lizzy}, \text{T.J.}, \text{Johnny}~\}\] Note that it doesn’t matter what order you list the members in. The set \(F\) of females contains Mom and Lizzy, but it’s not like Mom is the “first" female or anything. That doesn’t even make any sense. There is no “first." A set’s members are all equally members. So \(P\) is the same whether we write it like this: \[P = \{~\text{Dad}, \text{Mom}~\}\] or this: \[P = \{~\text{Mom}, \text{Dad}~\}.\] Those are just two different ways of writing the same thing.

The set \(E\) that had nobody in it can be written like this, of course: \[E = \{~\}\] but we sometimes use this special symbol instead: \[E = \varnothing.\] However you write it, this kind of set (one that has no elements) is referred to as an empty set.

The set \(H\), above, contained all the members of the group under consideration. Sometimes we’ll refer to “the group under consideration" as the “domain of discourse." It too is a set, and we usually use the symbol \(\Omega\) to refer to it.¹ So in this case, \[\Omega = \{~\text{Mom}, \text{Johnny}, \text{T.J.}, \text{Dad}, \text{Lizzy}~\}.\] Another symbol we’ll use a lot is “\(\in\)", which means “is a member of." Since Lizzy is a female, we can write: \[\text{Lizzy} \in F\] to show that Lizzy is a member of the \(F\) set. Conversely, we write: \[\text{T.J.} \notin F\] to show that T.J. is not.

As an aside, I mentioned that every item is either in, or not in, a set: there are no shades of gray. Interestingly, researchers have developed another body of mathematics called (I kid you not) “fuzzy set theory." Fuzzy sets change this membership assumption: items can indeed be “partially in" a set. One could declare, for instance, that Dad is “10% female," which means he’s only 10% in the \(F\) set. That might not make much sense for gender, but you can imagine that if we defined a set \(T\) of “the tall people," such a notion might be useful. At any rate, this example illustrates a larger principle which is important to understand: in math, things are the way they are simply because we’ve decided it’s useful to think of them that way. If we decide there’s a different useful way to think about them, we can define new assumptions and proceed according to new rules. It doesn’t make any sense to say “sets are (or aren’t) really fuzzy": because there is no “really." All mathematics proceeds from whatever mathematicians have decided is useful to define, and any of it can be changed at any time if we see fit.

Some authors use the symbol U for this, and call it the “universal set.”