Question

If U={1,2,3,4,5,6,7,8}, A={2,4,6,8} and B={2,4,8}. Then Prove that (A∩B)' = A'∪B'.​

Answer 1

Step-by-step explanation:

Given :-

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

A={2,4,6,8}

and B={2,4,8}.

To find :-

Prove that (A∩B)' = A'∪B'

Solution :-

Given sets are :

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

A={2,4,6,8}

and B={2,4,8}.

Finding (A∩B)':-

We know that

(A∩B)' = U -(A∩B)

A∩B = {2,4,6,8} ∩ { 2,4,8}

A∩B = {2,4,8}

Now

U -(A∩B)

=> {1,2,3,4,5,6,7,8} - { 2,4,8}

=> {1,3,5,6,7}

(A∩B)' = {1,3,5,6,7} ----------------------(1)

Finding A'∪B':-

We know that

A' = U - A

=> { 1,2,3,4,5,6,7,8} - { 2,4,6,8}

=> A' = { 1,3,5,7}

B' = U - B

=> { 1,2,3,4,5,6,7,8} - { 2,4,8}

=> B' = { 1,3,5,6,7}

Now,

A' U B'

=> { 1,3,5, 7} U { 1,3,5,6,7}

=> A'UB' = { 1,3,5,6,7} ----------------------(2)

From (1) &(2)

(A∩B)' = A'∪B'

Hence, Proved.

Answer:-

(A∩B)' = A'∪B'

Used formulae:-

• A' = U - A

• U is the universal set

• The set of elements in either A or in B or in both is called AUB .

• The set of Common elements in both A and B is called A∩B.

• The set of elements which does not belong to A contained in U is called A'.

• (A U B)' = A' ∩ B'

• (A∩B)' = A'∪B'.
• These are called De Morgan's laws on sets

Answer 2

Answer:

Given ,

• U ={1,2,3,4,5,6,7,8}
• A ={2,4,6,8}
• B ={2,4,8}

To prove : —

• (A∩B)' = A'∪B'.

Now,

A∩B = { 2 ,4 , 8}

So,

(A∩B)' = U - (A∩B)

= {1,2,3,4,5,6,7,8} - { 2 ,4 , 8}

= { 1 , 3 , 5 , 6 , 7 } ---------(1)

And

A' = U - A

= {1,2,3,4,5,6,7,8} - {2,4,6,8}

= { 1 , 3 , 5 , 7 }

B' = U - B

={1,2,3,4,5,6,7,8} - {2,4,8}

= { 1 , 3 , 5 , 6 , 7 }

So,

A'∪B' = { 1 , 3 , 5 , 7 } ∪ { 1 , 3 , 5 , 6 , 7 }

= { 1 , 3 , 5 , 6 , 7 } -----------(2)

From (1) and (2) ,

(A∩B)' = A'∪B'.

$\underline{♠ Proved ♠}$