Complement

What is Complement?

Someone asks: “Which students didn’t pass the test?”

To answer, you need two things:

1. The list of students who passed
2. The list of all students

Without knowing all the students, you can’t figure out who didn’t pass.

That’s the key idea: complement means “everything else” — but “everything else” only makes sense when you know what “everything” is.

The Universal Set

Before we can talk about complement, we need a universal set.

Symbol: $U$

The universal set is the “everything” we’re working with. It defines the boundaries of our world.

Examples:

The universal set changes depending on what you’re working with. When talking about weekdays, you don’t care about numbers. When talking about digits, you don’t care about letters.

The universal set sets the stage. Everything happens inside it.

The Complement

The complement of A is everything in the universal set that’s not in A.

Symbol: $A^c$ (also written $\overline{A}$ or $A'$)

Step-by-Step Example

Let’s say we’re working with days of the week.

$U = \{\text{Mon, Tue, Wed, Thu, Fri, Sat, Sun}\}$

$A = \{\text{Mon, Tue, Wed, Thu, Fri}\} \quad \text{(weekdays)}$

Find $A^c$:

Go through each day in U. Is it in A?

$A^c = \{\text{Sat, Sun}\} \quad \text{(the weekend)}$

If you’re not a weekday, you’re a weekend day. There’s no third option.

Properties

Together, they cover everything:

$A \cup A^c = U$

Think about it: every element in U is either in A, or not in A. There’s no third option. So when you combine A with “not A”, you get everything.

Example: Weekdays ∪ Weekends = All days of the week.

They never overlap:

$A \cap A^c = \emptyset$

Nothing can be both in A and not in A at the same time. That would be a contradiction.

Example: A day can’t be both a weekday and a weekend day.

Double complement:

$(A^c)^c = A$

The complement of “not A” is… A. Double negation cancels out.

Example: “Not a non-weekday” = weekday.

Complement of the universe:

$U^c = \emptyset$

Nothing is outside of everything.

Complement of empty set:

$\emptyset^c = U$

Everything is outside of nothing.

Connection to Set Difference

Complement is just set difference with U:

$A^c = U \setminus A$

“Everything except A” is the same as “the universe minus A.”

This is why complement requires a universal set — without U, you can’t compute the difference.

De Morgan’s Laws

These are two powerful rules about how complement interacts with union and intersection.

First law:

$(A \cup B)^c = A^c \cap B^c$

In words: “Not in A-or-B” is the same as “not in A and not in B.”

Example: You’re not in the chess club or drama club. That means you’re not in chess and you’re not in drama.

Second law:

$(A \cap B)^c = A^c \cup B^c$

In words: “Not in A-and-B” is the same as “not in A or not in B.”

Example: You’re not taking both math and physics. That means you’re missing at least one — either not taking math, or not taking physics (or neither).

De Morgan’s Laws: When you complement, $\cup$ becomes $\cap$ and $\cap$ becomes $\cup$.

The Formal Definition

$A^c = \{x \mid x \in U \text{ and } x \notin A\}$

This reads: “The set of all $x$ such that $x$ is in the universe and $x$ is not in A.”

Complement answers: “What’s left?”