# Complement of a set

## Complement of a Set Definition

The Complement of a set $A$ may be defined as the set of all the elements that are present in its Universal Set $U$ but not in the set $A$.

To understand the concept of Complement of a Set, let us consider a set $A=\{1, 2, 3, 4, 5\}$ and let the Universal Set of $A$ be $U=\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$.

The Complement of Set $A$ will have all the elements that are present in the Universal Set, $U$ but not in $A$. In this case, $6, 7, 8$ and $9$ are such elements. Therefore, the Complement of Set $A$ will have the elements $6, 7, 8$ and $9$.

## Set Builder Notation for Complement of a Set

The Set Builder Notation for the Complement of a Set is given by,

$\displaystyle {A}'=[{x:x\in U,x\notin A}]$

## Complement of a set Symbol

The Symbol for representing the Complement of a Set $A$ is given by ${{A}'}$ or sometimes also given by ${{A}^{C}}$.

## Complement of a Set Examples

Let us consider a set $A = \{1, 2, 3\}$ and its universal set $B$ is

$B = \{1, 2, 3, 4, 5, 6, 7\}$

The above figure shows the Venn diagram of the Complement of a Set. Here, the Universal Set $B$ has some extra elements i.e, $4, 5, 6$ and $7$ that are not present in the set $A$, shown by the green shade in the above figure. The set of these elements are the Complement of the Set $A$.

Therefore,

$\displaystyle {{A}'} = \{4, 5, 6, 7\}$

## Properties of Complement of a Set

• The Union of set $A$ with its Complement $\displaystyle {{A}'}$ will give the universal set.

$\displaystyle A\cup {A}'=U$

• The Intersection of set $A$ with its Complement $\displaystyle {{A}'}$ will result an Empty Set.

$\displaystyle A\cap {A}'=\phi$

• The Complement of an Empty set $\displaystyle \phi$ is the universal set $U$.

$\displaystyle {\phi }'=U$

• The Complement of an Universal set $U$ is an Empty set.

$\displaystyle {U}'=\phi$

• If $\displaystyle A\subseteq B$ then $\displaystyle {B}'\subseteq {A}'$
• The Double Complement of set $A$ is the set $A$ itself.

$\displaystyle ({A}'{)}'=A$

## How to Find the Complement of a Set?

Step 1: Let $A=\{1, 2, 3\}$

Step 2: Let the universal set of $A$ is $U=\{1, 2, 3, 4, 5, 6\}$

Step 3: Find the elements that are present in the universal set $U$ but not in $A$.

The elements are: $\{4, 5, 6\}$

Step 4: The above elements found in Step 3 are the complement of set $A$.

Complement of $A$, ${A}' = \{4, 5, 6\}$

### What is Complement of a set?

The Complement of a set A may be defined as the set of all the elements that are present in its Universal Set U but not in set A.

### How do you find the complement of a set?

Step 1: Let A={1, 2, 3}

Step 2: Let the universal set of A is U={1, 2, 3, 4, 5, 6}

Step 3: Find the elements that are present in the universal set U but not in A.

The elements are: {4, 5, 6}

Step 4: The above elements found in Step 3 are the complement of set A.

Complement of A, A' = {4, 5, 6}

### What do you mean by the complement of a Set?

The complement of a set is the collection of all the elements that are present in its universal set but not in the set.

### What are the properties of the Complement of a set?

The Properties of Complement of a Set are:

1. The union of set A with its complement A' will give the universal set.
2. The intersection of set A with its complement A' will result in an Empty Set.
3. The complement of an Empty set phi is the universal set U.
4. The complement of a universal set U is an Empty set.