Definition: Set

Definition: A set is a collection of elements.

Notation: It is often convenient to name our sets so that we don’t have to keep referring to them by their elements or how they’re defined. Thus, if you hand me a set I can just call it “A”, and then we’ll both know what I’m talking about when I refer to the set A. We would like to keep track of what elements are in A, however, and so we use the following bracket notation for the elements in a set. If A is a set of apples which we’ve labeled “apple1”, “apple2”, and “apple3”, then we write $A=\{\mathrm{apple1}, \mathrm{apple2}, \mathrm{apple3}\}$ . Similarly, A could be a set of numbers, perhaps 1, 2, 3, 4, and 5, in which case we write $A=\{1, 2, 3, 4, 5\}$ .

The order in which we write the elements in the brackets does not matter. Thus, $\{1, 2, 3, 4, 5\}=\{2, 4, 1, 5, 3\}$ . However, sometimes we use ordering to imply what the elements in a set are. Thus, if A is the set of natural numbers, then $A=\{1, 2, 3, 4, ...\}$ where we use “…” to imply that the ordering that we’ve used goes on forever. We need such an “implied notation” because we can’t explicitly write out all of the elements when there are infinitely many of them!

Examples:

10 apples form a set of apples, and the number of elements in the set is 10. (Trivial example)
The natural numbers form a set of numbers (as we just saw), and the number of elements in this set is infinite.
The set {apple, donkey, this glass of orange juice, democracy, Kobe Bryant} forms a set of 5 elements.
5 sets of 2 apples forms a set of 5 sets. Think about this one for a second. The enemy gives you 5 sets, each with 2 apples (or equivalently, 2 elements) in it. I can then form 1 set that has 5 elements in it, where each element is a set of 2 apples. Thus, your 1 set has 5 elements even though you have 10 apples! For more on sets of sets, take a look at lesson 4.

Notes:

As we saw above, the elements of a set can be (virtually) anything, even other sets!
A set can be finite in size, or infinite (we also saw this above).
We can’t be too relaxed in what we call a set. If we try to define “the set of all sets”, we quickly run into problems, as can be discussed in the post on Russell’s Paradox (lesson 5).

A fun way to spend your time: try to define some really whacky sets and then ask yourself questions about them. Are they finite or infinite? How many different subsets does it have? What kinds of functions can you define between two of them, or from a set to itself?

We first introduce sets in lesson 2, and continue studying them pretty much throughout the entirety of this site!

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