In mathematics, we often need to group things together. Think about your favorite songs, the planets in our solar system, or the vowels in the alphabet. Each of these is a collection of specific, distinct items. This simple idea of a collection is one of the most fundamental concepts in math.

We call these collections sets. A set is a collection of distinct objects, where each object is called an element or a member of the set. Two key ideas here are "well-defined" and "distinct."

• Well-defined means we can tell for certain whether an object belongs in the set or not. The "set of tall people" isn't well-defined because "tall" is subjective. But the "set of people over six feet tall" is well-defined.
• Distinct means that each element in a set is unique. We don't list the same item more than once.

To write a set, we use curly braces {} and list the elements inside, separated by commas. We usually give sets a name using a capital letter.

For example, if we want to define a set V that contains the vowels of the English alphabet, we would write it like this:

Here's another example. Let's create a set N of the first five positive whole numbers:

There are two important rules for writing sets.

Once we have a set, we often want to talk about whether something is in it. We use special symbols for this.

The symbol $∈$ means "is an element of." The symbol $∉$ means "is not an element of."

Let's use our set of vowels, $V = \{a, e, i, o, u\}$. We can say that 'i' is an element of V and 'b' is not.

Here's how it works with our number set, $N = \{1, 2, 3, 4, 5\}$:

This simple notation helps us be precise about what belongs in our collections.

Ready to check your understanding?

And that's the basic idea of a set. It's a simple but powerful tool for organizing information, and it's the starting point for many other ideas in mathematics.