Let A = {x: x ∈ N}, B = {x: x = 2n, n ∈ N), C = {x: x = 2n –

1.	Let A = {x: x ∈ N}, B = {x: x = 2n, n ∈ N), C = {x: x = 2n – 1, n ∈ N} and, D = {x: x is a prime natural number} Find:(i) A ∩ B(ii) A ∩ C(iii) A ∩ D(iv) B ∩ C(v) B ∩ D(vi) C ∩ D
Answer» A = All the natural numbers i.e. {1, 2, 3…..} B = All the even natural numbers i.e. {2, 4, 6, 8…} C = All the odd natural numbers i.e. {1, 3, 5, 7……} D = All the prime natural numbers i.e. {1, 2, 3, 5, 7, 11, …} (i) A ∩ B Here, A contains all elements of B. ∴ B ⊂ A = {2, 4, 6, 8…} ∴ A ∩ B = B (ii) A ∩ C Here, A contains all elements of C. ∴ C ⊂ A = {1, 3, 5…} ∴ A ∩ C = C (iii) A ∩ D Here, A contains all elements of D. ∴ D ⊂ A = {2, 3, 5, 7..} ∴ A ∩ D = D (iv) B ∩ C B ∩ C = ϕ Since, there is no natural number which is both even and odd at same time. (v) B ∩ D B ∩ D = 2 Since, {2} is the only natural number which is even and a prime number. (vi) C ∩ D C ∩ D = {1, 3, 5, 7…} = D – {2} Hence, every prime number is odd except {2}.

Let A = {x: x ∈ N}, B = {x: x = 2n, n ∈ N), C = {x: x = 2n – 1, n ∈ N} and, D = {x: x is a prime natural number} Find:(i) A ∩ B(ii) A ∩ C(iii) A ∩ D(iv) B ∩ C(v) B ∩ D(vi) C ∩ D

Answer»

A = All the natural numbers i.e. {1, 2, 3…..}

B = All the even natural numbers i.e. {2, 4, 6, 8…}

C = All the odd natural numbers i.e. {1, 3, 5, 7……}

D = All the prime natural numbers i.e. {1, 2, 3, 5, 7, 11, …}

(i) A ∩ B

Here, A contains all elements of B.

∴ B ⊂ A = {2, 4, 6, 8…}

∴ A ∩ B = B

(ii) A ∩ C

Here, A contains all elements of C.

∴ C ⊂ A = {1, 3, 5…}

∴ A ∩ C = C

(iii) A ∩ D

Here, A contains all elements of D.

∴ D ⊂ A = {2, 3, 5, 7..}

∴ A ∩ D = D

(iv) B ∩ C

B ∩ C = ϕ

Since, there is no natural number which is both even and odd at same time.

(v) B ∩ D

B ∩ D = 2

Since, {2} is the only natural number which is even and a prime number.

(vi) C ∩ D

C ∩ D = {1, 3, 5, 7…}

= D – {2}

Hence, every prime number is odd except {2}.