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## 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\}

## Frequently Asked Questions (FAQ)

### 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:

- 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.

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