Question

U = {x | x Î N, 1 < x < 10},A = {1, 3, 5, 7, 9} and B = {2, 5, 8} Verify the De morgan’s Law.​

Answer 1

Step-by-step explanation:

Given :-

U = {x | x Î N, 1 < x < 10}

A = {1, 3, 5, 7, 9}

B = {2, 5, 8}

To find :-

Verify the De morgan’s Law.

Solution :-

Given sets are :

U = {x | x Î N, 1 < x < 10}

=> U = { 2,3,4,5,6,7,8,9}

A = {1, 3, 5, 7, 9}

B = {2, 5, 8}

We know that

( A ∪ B)' = A' ∩ B' is called De morgan’s Law.

Finding ( A ∪ B)' :-

A U B = {1, 3, 5, 7, 9} U {2, 5, 8}

A U B = { 1,2,3,5,7,8,9 }

We know that

( A ∪ B)' = U - ( A ∪ B)

=> ( A ∪ B)' = {2,3,4,5,6,7,8,9} - {1,2,3,5,7,8,9 }

=> ( A ∪ B)' = { 4,6 } -----------(1)

Finding A' ∩ B':-

We know that

A' = U - A

=> A' = {2,3,4,5,6,7,8,9} - {1, 3, 5, 7, 9}

=> A' = {2,4,6,8}

B' = U - B

=> B' = {2,3,4,5,6,7,8,9} - {2, 5, 8}

=> B' = { 3,4,6,7,9}

Now ,

A' ∩ B'

=> {2,4,6,8} ∩ {3,4,6,7,9}

=> { 4,6}

A' ∩ B' = { 4,6} -------------(2)

From (1) & (2)

( A ∪ B)' = A' ∩ B'

Verified the given relation .

Used formulae:-

• De morgan’s Law is ( A ∪ B)' = A' ∩ B'

• A U B is the set of elements is either in A or in B or in both.

• A U B ={x:x€A or x€B}

• U is the universal set.

• A' is the set of all elements in U which are not belongs to A.

• A' = U-A