In math, a set is simply a collection of distinct objects. Think of it like a box where you keep your favorite things. The objects inside are called the elements or members of the set.

You encounter sets all the time. Your group of friends is a set. The collection of apps on your phone is a set. The days of the week form a set: {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}.

To talk about sets, we need a clear way to write them. We use curly braces {} to enclose the elements. There are two main ways to describe what's inside.

For example, the set of the first five positive whole numbers is:

$S = \{1, 2, 3, 4, 5\}$

The order of elements doesn't matter. The set $\{1, 2, 3\}$ is the same as $\{3, 1, 2\}$. Also, remember that elements must be distinct. The collection $\{1, 2, 2, 3\}$ is just the set $\{1, 2, 3\}$.

But what if we want to describe a large set, like all even numbers? Listing them all would be impossible. For that, we use set-builder notation.

Let's look at the set of vowels again. In set-builder notation, we could write it like this:

You can read this as: "V is the set of all elements x such that x is a vowel in the English alphabet." The vertical bar | means "such that." It separates the variable from the rule that defines the elements.

Two particular types of sets come up so often that they get their own names and symbols.

We represent the empty set with the symbol $\emptyset$ or just empty curly braces {}.

The Universal Set is the set of all possible elements for the problem we're considering. It's the big picture, or the context. If we are discussing single-digit numbers, the universal set, often labeled $U$, would be:

In this diagram, the universal set $U$ is all integers from 1 to 10. The set $A$ is the set of even numbers within that universe. All other sets we might discuss in this context would also draw their elements from $U$.

Understanding what a set is and how to write one is the first step. It gives us a powerful language to group and describe mathematical objects, which is a key skill you'll use in more advanced topics.