Mathematics Complement of a Set and Some Properties of Complement Sets

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Class 11 Chapter 1 - SETS

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

`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`

`\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)

(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'`.

2 min Review Quiz

2 min Review Quiz

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