Question

1. If A and B are any two sets then prove that :
(i) (AUB)’=A’∩B’ (ii) (A∩B)’=A’UB’

Answer 1

Proofs

1)

To prove

(A∪B)' =A'∩B'

let

    x ∈ (A∪B)'

⇒  x ∉ (A∪B)

⇒  x ∉ A and x ∉ B

⇒  x ∈ A' and x ∈ B'

⇒  x ∈ A'∩B'

⇒ (A∪B)' ⊂ A'∩B'   ......(1)

Now

Let

   y ∈ A'∩B'

⇒ y ∈ A' and y ∈ B'

⇒ y ∉ A and y ∉ B

⇒ y ∉ (A∪B)

⇒ y ∈ (A∪B)'

⇒ A'∩B' ⊂ (A∪B)'    ........(2)

From (1) and (2)

(A∪B)' = A'∩B'

Hence prove

2)

To prove

(A∩B)' = A'∪B'

let

    x ∉ (A∩B)'

⇒  x ∈ (A∩B)

⇒  x ∈ A and x ∈ B

⇒  x ∉ A' and x ∉ B'

⇒  x ∉ A'∪B' .......(1)

It show the element which not present in (A∩B)' also not present in A'∪B'

Now

Let

   y ∉ A'∪B'

⇒ y ∉ A' and y ∉ B'

⇒ y ∈ A and y ∈ B

⇒ y ∈ (A∪B)

⇒ y ∉ (A∪B)'       ......(2)

It show the element which not present in A'∪B' also not present in (A∩B)'

Thus from (1) and (2) it clear only those element present in  A'∪B' which is also present  (A∩B)' and vise versa

Thus

(A∩B)' = A'∪B'

Hence prove.

Answer 2

Answer:

HOPE THIS HELPS U

Step-by-step explanation:

1)To prove

(A∪B)' =A'∩B'

let

x ∈ (A∪B)'

⇒  x ∉ (A∪B)

⇒  x ∉ A and x ∉ B

⇒  x ∈ A' and x ∈ B'

⇒  x ∈ A'∩B'

⇒ (A∪B)' ⊂ A'∩B'   ......(1)

Now

Let

 y ∈ A'∩B'

⇒ y ∈ A' and y ∈ B'

⇒ y ∉ A and y ∉ B

⇒ y ∉ (A∪B)

⇒ y ∈ (A∪B)'

⇒ A'∩B' ⊂ (A∪B)'    ........(2)

From (1) and (2)

(A∪B)' = A'∩B'

Hence prove

2)To prove

(A∩B)' = A'∪B'

let

x ∉ (A∩B)'

⇒  x ∈ (A∩B)

⇒  x ∈ A and x ∈ B

⇒  x ∉ A' and x ∉ B'

⇒  x ∉ A'∪B' .......(1)

it show the element which not present in (A∩B)' also not present in A'∪B'

Now

Let

 y ∉ A'∪B'

⇒ y ∉ A' and y ∉ B'

⇒ y ∈ A and y ∈ B

⇒ y ∈ (A∪B)

⇒ y ∉ (A∪B)'       ......(2)

It show the element which not present in A'∪B' also not present in (A∩B)'

Thus from (1) and (2) it clear only those element present in  A'∪B' which is also present  (A∩B)' and vise versa

Thus

(A∩B)' = A'∪B'

Hence proved.........