Question

8)
Verify De Morgan's law for difference, if
U = {natural numbers less than 10),
A = {2, 1,4,5), and
B = {odd number less than 10}.( please send the answer) please ​

Answer 1

Given :

• U = {natural numbers less than 10}
• A = {2, 1,4,5}
• B = {odd number less than 10}

To Verify :

• De Morgan's law for difference

Solution :

U = {natural numbers less than 10}

• U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 }

A = {2, 1,4,5)

• A = {2, 1,4,5)

B = {odd number less than 10}

• B = { 1 , 3 , 5 , 7 , 9 }

So,

U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 }

A = { 2 , 1 , 4 , 5}

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

By using De Morgan's law,

(A ∪ B) ' = A ' ∩ B '

★ LHS = (A ∪ B) '

⇒ A ∪ B = { 2 , 1 , 4 , 5} ∪ { 1 , 3 , 5 , 7 , 9 }

⇒ A ∪ B = { 1 , 2 , 3 , 4 , 5 , 7 , 9 }

(A ∪ B) ' = { 6 , 8 }

LHS = { 6 , 8 }

★ RHS = A ' ∩ B '

⇒ A ' = { 3 , 6 , 7 , 8 , 9 }

⇒ B ' = { 2 , 4 , 6 , 8 }

A ' ∩ B ' = { 3 , 6 , 7 , 8 , 9 } ∩ { 2 , 4 , 6 , 8 }

A ' ∩ B ' = { 6 , 8 }

RHS = { 6 , 8 }

LHS = RHS

(A ∪ B) ' = A ' ∩ B '

•°• HENCE VERIFIED . . .

Answer 2

Solution -

We have three sets,

• U = {natural numbers less than 10}
• A = {2, 1, 4, 5}
• B = {odd number less than 10}

⠀

Firstly, we will write all the sets in roaster form

➝ U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

➝ A = {2, 1, 4, 5}

➝ B = {1, 3, 5, 7, 9}

⠀

Now, De Morgan's law states that

⠀⠀⠀⠀⠀⠀☆ (A ∪ B)' = A' ∩ B'

⠀

Firstly, let us find L.H.S

➝ A∪B = {2, 1, 4, 5} ∪ {1, 3, 5, 7, 9}

➝ A∪B = {1, 2, 3, 4, 5, 7, 9}

➝ (A∪B)' = {6, 8}

⠀

Similarly, we will find R.H.S

➝ A' = {3, 6, 7, 8, 9}

➝ B' = {2, 4, 6, 8}

➝ A' ∩ B' = {6, 8}

⠀

We can see that,

⇢ L.H.S = {6, 8}

⇢ R.H.S = {6, 8}

⇢ L.H.S = R.H.S

★ Hence proved ★