RD Chapter 1- Sets Ex-1.1 |
RD Chapter 1- Sets Ex-1.2 |
RD Chapter 1- Sets Ex-1.3 |
RD Chapter 1- Sets Ex-1.4 |
RD Chapter 1- Sets Ex-1.6 |
RD Chapter 1- Sets Ex-1.7 |
RD Chapter 1- Sets Ex-1.8 |

If A and B are two sets such that A ⊂ B, then Find:

(i) A ⋂ B

(ii) A ⋃ B

**Answer
1** :

(i) A ∩ B

A ∩ B denotes A intersection B. Common elements of A and B consists in this set.

Given A ⊂ B, every element of A are already an element of B.

∴ A ∩ B = A

(ii) A ⋃ B

A ∪ B denotes A union B. Elements of either A or B or in both A and B consist in this set.

Given A ⊂ B, B is having all elements including elements of A.

∴ A ∪ B = B

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find:

(i) A ∪ B

(ii) A ∪ C

(iii) B ∪ C

(iv) B ∪ D

(v) A ∪ B ∪ C

(vi) A ∪ B ∪ D

(vii) B ∪ C ∪ D

(viii) A ∩ (B ∪ C)

(ix) (A ∩ B) ∩ (B ∩ C)

(x) (A ∪ D) ∩ (B ∪ C).

**Answer
2** :

In general X ∪ Y = {a: a ∈ X or a ∈ Y}

X ∩ Y = {a: a ∈ X and a ∈ Y}.

(i) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

A ∪ B = {x: x ∈ A or x ∈ B}

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

(ii) A = {1, 2, 3, 4, 5}

C = {7, 8, 9, 10, 11}

A ∪ C = {x: x ∈ A or x ∈ C}

= {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}

(iii) B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

B ∪ C = {x: x ∈ B or x ∈ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

(iv) B = {4, 5, 6, 7, 8}

D = {10, 11, 12, 13, 14}

B ∪ D = {x: x ∈ B or x ∈ D}

= {4, 5, 6, 7, 8, 10, 11, 12, 13, 14}

(v) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

A ∪ B = {x: x ∈ A or x ∈ B}

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

A ∪ B ∪ C = {x: x ∈ A ∪ B or x ∈ C}

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

(vi) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

D = {10, 11, 12, 13, 14}

A ∪ B = {x: x ∈ A or x ∈ B}

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

A ∪ B ∪ D = {x: x ∈ A ∪ B or x ∈ D}

= {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14}

(vii) B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

D = {10, 11, 12, 13, 14}

B ∪ C = {x: x ∈ B or x ∈ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

B ∪ C ∪ D = {x: x ∈ B ∪ C or x ∈ D}

= {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

(viii) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

B ∪ C = {x: x ∈ B or x ∈ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

A ∩ B ∪ C = {x: x ∈ A and x ∈ B ∪ C}

= {4, 5}

(ix) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

(A ∩ B) = {x: x ∈ A and x ∈ B}

= {4, 5}

(B ∩ C) = {x: x ∈ B and x ∈ C}

= {7, 8}

(A ∩ B) ∩ (B ∩ C) = {x: x ∈ (A ∩ B) and x ∈ (B ∩ C)}

= ϕ

(x) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

D = {10, 11, 12, 13, 14}

A ∪ D = {x: x ∈ A or x ∈ D}

= {1, 2, 3, 4, 5, 10, 11, 12, 13, 14}

B ∪ C = {x: x ∈ B or x ∈ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

(A ∪ D) ∩ (B ∪ C) = {x: x ∈ (A ∪ D) and x ∈ (B ∪ C)}

= {4, 5, 10, 11}

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
3** :

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

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

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

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

(i) A ∩ B

A contains all elements of B.

∴ B ⊂ A = {2, 4, 6, 8…}

∴ A ∩ B = B

(ii) A ∩ C

A contains all elements of C.

∴ C ⊂ A = {1, 3, 5…}

∴ A ∩ C = C

(iii) A ∩ D

A contains all elements of D.

∴ D ⊂ A = {2, 3, 5, 7..}

∴ A ∩ D = D

(iv) B ∩ C

B ∩ C = ϕ

There is no natural number which is both even and odd at same time.

(v) B ∩ D

B ∩ D = 2

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

(vi) C ∩ D

C ∩ D = {1, 3, 5, 7…}

= D – {2}

Every prime number is odd except {2}.

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