Question

let u = {1,2,3,4,5,6} ,a= {2,3} and b ={3,4,5} find a' , b' , a' intersection to b' , a union b and hence show that (a union b )'= a' union b'​

Answer 1

Step-by-step explanation:

Given:-

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

A= {2,3} ,

and B={3,4,5} .

To find:-

Find the following:-

1)A'

2)B'

3) A' n B'

4) A' U B'

5) Show that (AUB)' = A' U B'

Solution:-

Given sets are :

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

A= {2,3} ,

and B={3,4,5} .

1) A' = U - A

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

=> { 1,4,5,6}

A' = {1,4,5,6}

2) B' = U - B

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

=> {1,2,6}

B' = {1,2,6}

3) A' n B':-

= { 1,4,5,6} n { 1,2,6}

= {1,6}

A' n B' = { 1,6 } -------(1)

4) A' U B':

={1,4,5,6} U { 1,6}

= { 1,4,5,6}

A' U B' = { 1,4,5,6}-----(2)

5)(AUB)' = A' U B':

AUB =

{ 2,3} U { 3,4,5}

= { 2,3,4,5}

LHS:-

(AUB)'

=>U - (AUB)

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

=> {1,6}

(AUB)' = { 1,6}

LHS= {1,6}---------(3)

A' U B' = { 1,4,5,6}

RHS = {1,4,5,6} ----(4)

From (3)&(4)

(AUB)' ≠ A' U B'

But

From (1)&(3)

(AUB)' = A' n B'

Answer:-

1)A' = { 1,4,5,6}

2)B' = {1,2,6}

3) A' n B' = { 1,6 }

4) A' U B' = { 1,4,5,6}

5)(AUB)' = A' n B'