Complement of a Set

Complement of a Set - Definition, Explanation, and Examples

The complement of a set contains all elements that are not in the set but belong to a larger, universal set. It helps identify what is "outside" a given set within a specified context.

Definition of Complement

Let $U$ be the universal set, which contains all elements under consideration. The complement of a set $A$, denoted by $A^{\prime }$ or $\bar{A}$, is defined as:

$A^{\prime }=\{x\in U:x\notin A\}$

In words, the complement of $A$ is the set of all elements in the universal set $U$ that are not in $A$.

Understanding Complement

The complement tells us everything outside the set $A$ within the universe of discourse.

Step-by-Step Explanation

1. Identify the universal set $U$. 2. Identify the elements of the set $A$. 3. Find all elements in $U$ that are not in $A$. 4. Collect these elements to form the complement set $A^{\prime }$.

Examples of Complement of a Set

Example 1: Numbers

Let the universal set be:

$U=\{1,2,3,4,5,6,7,8,9,10\}$

and let

$A=\{2,4,6,8,10\}$

Step 1: Universal set $U$ contains numbers 1 to 10. Step 2: Set $A$ contains even numbers 2, 4, 6, 8, 10. Step 3: Elements in $U$ but not in $A$ are odd numbers: 1, 3, 5, 7, 9. Step 4: Complement of $A$ is:

$A^{\prime }=\{1,3,5,7,9\}$

Example 2: Letters

Let

$U=\{\text{a},\text{b},\text{c},\text{d},\text{e}\}$

and

$B=\{\text{a},\text{c},\text{e}\}$

Then,

$B^{\prime }=\{\text{b},\text{d}\}$

since these are the letters in $U$ not in $B$.

Example 3: Shapes

Consider a universal set of shapes:

$U=\{\text{circle},\text{square},\text{triangle},\text{rectangle}\}$

and set

$C=\{\text{circle},\text{triangle}\}$

The complement of $C$ is:

$C^{\prime }=\{\text{square},\text{rectangle}\}$

Example 4: Numbers Between 1 and 15

Let

$U=\{1,2,3,\dots ,15\}$

and

$D=\{5,6,7,8,9\}$

The complement of $D$ is all numbers from 1 to 15 except 5 through 9:

$D^{\prime }=\{1,2,3,4,10,11,12,13,14,15\}$

Example 5: Prime Numbers Up to 20

Let

$U=\{1,2,3,\dots ,20\}$

and

$E=\{2,3,5,7,11,13,17,19\}$ (the prime numbers)

Then the complement of $E$ is all numbers from 1 to 20 that are not prime:

$E^{\prime }=\{1,4,6,8,9,10,12,14,15,16,18,20\}$

Important Notes

• The complement depends on the universal set $U$.
• The union of a set and its complement is the universal set: $A\cup A^{\prime }=U$.
• The intersection of a set and its complement is the empty set: $A\cap A^{\prime }=\mathrm{\varnothing }$.

Summary

The complement of a set shows all elements outside the set within a defined universal set. It is a useful concept for understanding what is excluded from a set.