Sets and membership

Set theory is all about sets. For now, we can say that sets are some specific collections of things. These things can be of any sort, including sets themselves. Going back to file systems, in some important aspects, sets are like folders, which may contain other folders, which in turn may contain other folders, and so on. In fact, sets are denoted just as we denoted folders in our language $\mathcal{L}_f$; using curly braces. Thus, the following is a set: $$\{\text{Peter}, \text{Leo}, \text{Taylor}, \text{Curtis}\}$$

Sets have members. The members of a set are those things that are in the set. So the members of the set $\{\text{Peter}, \text{Leo}, \text{Taylor}, \text{Curtis}\}$ are Peter, Leo, Taylor, and Curtis. Set membership is denoted by $\in$. So for example, $$\text{Peter} \in \{\text{Peter}, \text{Leo}, \text{Taylor}, \text{Curtis}\}$$ We can denote sets by using capital letters. The first choice is usually $S$ (you can guess why…). This makes it easier to talk about them. So for example: $$\begin{gathered} S=\{\text{Peter}, \text{Leo}, \text{Taylor}, \text{Curtis}\}\\ \text{Peter} \in S \end{gathered}$$ Similarly, we can denote something not being in the set by using $\notin$. So for example: $$\text{Henry} \notin S$$ So far, this is rather simple. Now let’s look at some complications.

Sets are extensional entities. This is a fancy word to say that all a set depends on is what its members are. So in a set, there is no ordering between its members, and it doesn’t matter whether the set is represented as having some members twice (or however many times). Each member is only counted once, in whatever order. So for example: $$\begin{aligned} \{\text{Peter}, \text{Leo}, \text{Taylor}, \text{Curtis}\}&=\{\text{Peter}, \text{Peter}, \text{Leo}, \text{Taylor}, \text{Curtis}\}\\ \{\text{Peter}, \text{Leo}, \text{Taylor}, \text{Curtis}\}&= \{\text{Curtis}, \text{Leo}, \text{Peter}, \text{Taylor}\} \end{aligned}$$

There is one set that is unlike others, denoted $\emptyset$. This is the empty set. The empty set is so-called because it is, you guessed it, empty. In other words, it has no members. So for any candidate member $x$, $x \notin \emptyset$.

Now sets can be members of other sets, and it is very important to be clear whether something is in a set, or it is in another set that is part of another set, and so on. Note: since $\emptyset$ is a set, it can also be a member of other sets. Let’s look at an example: $$\begin{aligned} S&=\{a, b, c, \{a, d\}, d, \emptyset\}\\ Q&=\{\{a, b, c\}, \{d\}, d, \emptyset\} \end{aligned}$$

Now $S$ and $Q$ are not the same set. They do share some elements, but they differ in others. Since they do not have the same elements, they are not the same set. In particular, $\emptyset \in S, Q$, and $d \in S, Q$. But note that $a, b, c \notin Q$. What is in $Q$ is the set {a, b, c}. Conversely, $a, b, c \in S$, but not $\{a, b, c\}$. Similarly, $\{d\} \in Q$ but $\{d\} \notin S$, though again, $d \in S, Q$.