An Introduction to Sets in Mathematics

A set is where math begins. You’ll run into sets all the time when learning about functions, linear algebra, and plenty of other topics. A set is simply a collection of distinct objects. These objects, called elements, can be anything: numbers, people, letters, or even other sets. The tricky part for many learners is the notation, which can feel like a barrier.

We usually write a set by listing its elements inside curly brackets, like this:

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

This is read as “set A contains the numbers 1, 2, 3, 4, and 5.”

Everyday Examples of Sets

In your daily life, you are already familiar with collections of things. These are, in essence, everyday sets. Here are a few examples that you probably already know:

• Grocery list: {milk, bread, eggs}
• Colors of a rainbow: {red, orange, yellow, green, blue, indigo, violet}
• Students in a classroom: {Alice, Ben, Carlos, Dana}
• Coin toss outcomes: {head, tail}
• Dice outcomes: {1, 2, 3, 4, 5, 6}

Key Rules of Sets

As you can see from these examples, there are two key rules to remember about sets:

1. Order doesn’t matter. For instance, the set {milk, eggs, bread} is the exact same set as {bread, milk, eggs}.
2. Duplicates don’t count. In other words, the set {1, 1, 2} is just {1, 2}. Every element in a set must be unique.

Important Set Concepts and Symbols

With the basic idea of a set established, let’s look at some of the fundamental concepts and symbols you’ll encounter.

• Cardinality: This is just a fancy word for the size of a set. We denote it with vertical bars. For example, if A={1,2,3,4,5}, then the cardinality of A is ∣A∣=5.
• The Empty Set (∅): Furthermore, there is a special set that contains no elements at all. We call it the empty set, and its cardinality is zero, or ∣∅∣=0.
• Subsets (⊆): A set B is a subset of set A if every single element in B is also in A. For example, if A={1,2,3,4} and B={1,2}, then we can write B⊆A.

Special Sets in Mathematics

Beyond the simple examples we’ve seen, mathematicians often work with a few well-known sets that have special symbols:

• N: Natural numbers {1,2,3,…}
• Z: Integers {…,−2,−1,0,1,2,…}
• Q: Rational numbers (any number that can be written as a fraction, like 21​ or −5)
• R: Real numbers (all numbers on the number line, including irrationals like π or 2​)
• C: Complex numbers (numbers that involve i, where i2=−1)

Describing Sets with Rules

So far, we’ve defined sets by listing their elements. But what if you want to describe a set that’s too big to write out? For example, the set of all integers greater than 3. For this, we use set-builder notation:

$$B = \{x \in \mathbb{Z} \mid x > 3\}$$

This is a powerful way to define a set based on a rule. The symbol ∈ means “is an element of” or “belongs to.” Therefore, this expression reads as: “B is the set of all integers x such that x is greater than 3.”

Key Set Operations

Once you have a handle on what a set is, you can start combining them in useful ways using a few key operations.

1. Union (∪) The union of two sets is a new set that contains every element from either set. Example: {1,2,3}∪{3,4,5}={1,2,3,4,5}
2. Intersection (∩) The intersection of two sets is a new set containing only the elements that they have in common. Example: {1,2,3}∩{3,4,5}={3}
3. Difference (−) The difference of two sets is what’s in the first set but not the second. Example: {1,2,3}−{3,4,5}={1,2}

These operations mirror real life. For instance, if you and a friend each write down your favorite movies, the union is your complete combined watchlist, the intersection is the movies you both love, and the difference is your unique picks.