Mathematics Complement of a Set and Some Properties of Complement Sets
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### Topics Covered

star Complement of a Set
star Some Properties of Complement Sets

### Complement of a Set

\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"

• Let U be the universal set and A a subset of U. Then \color{blue} ul(\mathtt (THE \ \ COMPLEMENT \ \ OF\ \ A)) is the set of all elements of U which are not the elements of A.

• Symbolically, we write A′ to denote the complement of A with respect to U.

Thus, A′ = {x : x ∈ U "and" x ∉ A }. Obviously A′ = U – A

E.g. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. then
A′ = { 2, 4, 6, 8,10 }.

 \color{red} \ox \color{red} \mathbf(COMMON \ CONFUSION) :  If A is a subset of the universal set U, then its complement A′ is also a subset of U.

We have  \ \ \ \ A′ = { 2, 4, 6, 8, 10 }
Hence (A′ )′ = {x : x ∈ U "and" x ∉ A′}
 \ \ \ \ \ \ \ \ = {1, 3, 5, 7, 9} = A

Q 3077178086

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′.

Solution:

We note that 2, 4, 6, 8, 10 are the only elements of U which do not belong to
A. Hence A′ = { 2, 4, 6, 8,10 }.
Q 3027278181

Let U be universal set of all the students of Class XI of a coeducational school and A be the set of all girls in Class XI. Find A′.

Solution:

Since A is the set of all girls, A′ is clearly the set of all boys in the class.

### Some Properties of Complement Sets

(i). A ∪ A′ = U

(ii) A ∩ A′ = φ (Complement laws)

(iii) (A ∪ B)´ = A′ ∩ B′

(iv) (A ∩ B )′ = A′ ∪ B′ (De Morgan’s law)

(v) (A′ )′ = A (Law of double complementation )

(vi). φ′ = U and U′ = φ (Laws of empty set and universal set)
Q 3057278184

Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}.
Find A′, B′ , A′ ∩ B′, A ∪ B and hence show that ( A ∪ B )′ = A′ ∩ B′.

Solution:

Clearly A′ = {1, 4, 5, 6}, B′ = { 1, 2, 6 }. Hence A′ ∩ B′ = { 1, 6 } Also A ∪ B = { 2, 3, 4, 5 }, so that (A ∪ B )′ = { 1, 6 }
( A ∪ B )′ = { 1, 6 } = A′ ∩ B′
It can be shown that the above result is true in general. If A and B are any two subsets of the universal set U, then

( A ∪ B )′ = A′ ∩ B′. Similarly, ( A ∩ B )′ = A′ ∪ B′ . These two results are stated in words as follows :
Q 1925534461

Draw appropriate Venn diagram for each of the following :

(i) (A cup B)'

(ii) A' cap B'

(iii) (A cap B)'

(iv) A' cup B'
Class 11 Exercise 1.5 Q.No. 5
Solution:

(i) (A cup B)'

(ii) A' cap B'

(iii) (A cap B)'

(iv) A' cup B'

All shaded region formed by all horizontal and vertical lines is A' cup B'.