Let us consider a set

*X*as*X*= {2, 3, 6, 8}

Let us consider its universal set

*ξ*asξ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

**Can we find the set of elements in set**

**ξ****, which are not in**

**X****?**

We can find this by taking the difference of

*X*from*ξ*asξ −

*X*= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} − {2, 3, 6, 8} = {0, 1, 4, 5, 7, 9}This set (consisting of all the elements

*ξ*, which do not belong to*X*) is known as the complement of set*X*and we denote it by or.
Let X be any set and ξ be its universal set. The complement of set X is the set consisting of all the elements of ξ, which do not belong to X. It is denoted by X′ or X^{c} (read as complement of set X).Thus, X′ = {x|x∈ξ and x∉X} or X′ = ξ−X |

For the above sets

*X*and ξ, we may observe that*n*(*X*) = 4,*n*(ξ) = 10, and*n*() = 6.**Can we find any relation among them?**

We observe that:

n() = n (ξ) − n (X) |

This relation holds true for a set, its complement, and a universal set.

The other properties of a set and its complement are as follows.

(a)

(b)

(c)

(d)

(e)

(f) If

*X*⊆*Y*thenApart from these properties, there are two more properties for two sets

*A*and*B.*They are:(a)

(b)

These are also known as

**De Morgan’s laws**.Let us prove the first one.

Let

*A*= {1, 2, 3},*B*= {2, 3, 4}, and*ξ*= {1, 2, 3, 4, 5, 6}Now,

*A*∩*B*= {2, 3}Therefore, (

*A*∩*B*)′ = {1, 4, 5, 6}Also,

*A*′ = {4, 5, 6} and*B*′ = {1, 5, 6}∴

*A*′ ∪*B*′ = {1, 4, 5, 6}Clearly, we have

Similarly, we can prove the second one.

Now, how will we represent the complement of a set

*A*with the help of a Venn diagram?We know that if

*A*is a set and*ξ*is a universal set for the set*A*, then the complement of the set*A*is*A*^{c}=*ξ*−*A*.If we represent the sets

*ξ*and*A*by a Venn diagram, then we can easily represent*A*^{c}on it.For this, we represent the set

*A*by using a circle and*ξ*by using a rectangle (or a square which is bigger and encloses the circle). Now, the portion outside the set*A*, but inside the set*ξ*, represents the set*A*^{c}. This can be shown as follows:Let us look at some examples in order to understand these concepts better.

**Example 1:**

**If**

**A****and**

**B****are two sets and**

**ξ****is their universal set such that , and**

**n****(**

**B****) = 6, then how many elements are there in the complement of set**

**B****?**

**Solution:**

We know that,

We also know that,

Therefore, the complement of set

*B*contains 2 elements.**Example 2:**

**If**

**A****= {**

**x****, 1, 2, 3,**

**y****},**

**B****= {2, 4, 5,**

**y****}, and**

**ξ****= {**

**x****,**

**y****,**

**z****, 1, 2, 3, 4, 5, 6}, then show that**

**Solution:**

Now,

*A*^{c}=*ξ*−*A =*{*x, y, z,*1, 2, 3, 4, 5, 6} − {*x*, 1, 2, 3,*y*}= {*z*, 4, 5, 6}*B*

^{c }=

*ξ*−

*B*= {

*x, y*,

*z*, 1, 2, 3, 4, 5, 6} − {2, 4, 5,

*y*} = {

*x, y*, 1, 3, 6}

*A*∪

*B*= {

*x, y,*1, 2, 3, 4, 5}

∴(

*A*∪*B*)^{c}= {*z*, 6}Now,

*A*^{c}∩*B*^{c}= {*z*, 6}Clearly, (

*A*∪*B*)^{c}=*A*^{c}∩*B*^{c}**Example 3:**

**Find the following sets from the adjoining Venn-diagram.**

**(i) (**

**A**

**∩**

**B****)**

^{c}

**(ii)**

**A**^{c}

**(iii) (**

**A**

**∪**

**C****)**

^{c}

**(iv)**

**ξ****Solution:**

From the given Venn-diagram, we find that

(i) (

*A*∩*B*)^{c }*=*{1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19}(ii)

*A*^{c }= {3, 4, 6, 7, 9, 11, 12, 13, 14, 15}(iii) (

*A*∪*C*)^{c }= {3, 4, 7, 9, 11, 13}(iv)

*ξ*= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19}**Example 4:**

**Taking the set of first ten natural numbers as the universal set, find the set**

**(**

**B****−**

**A****)′ ∩**

**B****′, where**

**A****= {1, 2, 4, 9} and**

**B****= {2, 5, 7, 9, 8, 10, 1, 3}**

**Solution:**

*B*−

*A*= {5, 7, 8, 10, 3}

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

(

*B*−*A*) ′ = {1, 2, 4, 6, 9}*B*′ = {4, 6}

∴ (

*B*−*A*)′ ∩*B*′ = {4, 6}
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