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**Exercise 2.1**

**1. If **** find the values of **** and **

**Ans. **Here

and

and

and

and

**2. If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A **** B).**

**Ans. **Number of elements in set A = 3 and Number of elements in set B = 3

Number of elements in A B = 3 3 = 9

**3. If G = {7, 8} and H = {5, 4, 2}, find G **** H and H **** G.**

**Ans. **Given: G = {7, 8} and H = {5, 4, 2}

GH = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

And H G = {(5, 7), (4, 7), (2, 7), (5, 8), (4, 8), (2, 8)}

**4. State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly:**

**(i) If P = **** and Q = **** then P **** Q = **

**(ii) If A and B are non-empty sets, then A **** B is a non-empty set of ordered pairs **** such that **** A and **** B.**

**(iii) If A = {1, 2}, B = {3, 4}, then **

**Ans. (i)** Here P = and Q =

Number of elements in set P = 2 and Number of elements in set Q = 2

Number of elements in P Q = 2 2 = 4

But PQ = and here number of elements in P Q = 2

Therefore, statement is false.

**(ii)** True

**(iii)** True

**5. If A = **** find A **** A **** A.**

**Ans. **Here A =

A A =

A A A =

**6. If A **** B = **** find A and B.**

**Ans. **Given: A B =

A = set of first elements = and B = set of second elements =

**7. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that:**

**(i) **

**(ii) A **** C is a subset of B **** D.**

**Ans. **Given: A = {1, 2}, B = {1, 2, 3, 4}, C

= {5, 6} and D = {5, 6, 7, 8}

**(i)** = {1, 2, 3, 4} {5, 6} =

……….(i)

A B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}

A C = {(1, 5), (1, 6), (2, 5), (2, 6)

(AB) (A C) = ……….(ii)

Therefore, from eq. (i) and (ii),

= (A B) (A C)

**(ii)** A C = {(1, 5), (1, 6), (2, 5), (2, 6)

B D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8),

(4, 5), (4, 6), (4, 7), (4, 8),

Therefore, it is clear that each element of A C is present in B D.

A C B D

**8. Let A = {1, 2} and B = {3, 4}, write A B. How many sub sets will A B have? List them.**

**Ans. **Given: A = {1, 2} and B = {3, 4}

A B = {(1, 3), (1, 4), (2, 3), (2, 4)}

Number of elements in A B = 4

Therefore, Number of subsets of AB = = 16

The subsets are:

{(1, 3)}, {(1, 4)}, {(2, 3)}, {(2, 4)}, {(1, 3), (1, 4)}, {(1, 3), (2, 3)}, {(1, 3), (2, 4)}, {(1, 4), (2, 3)}

{(1, 4), (2, 4)}, {(2, 3), (2, 4)}, {(1, 3), (1, 4), (2, 3)}, {(1, 3), (1, 4), (2, 4)}, {(1, 3), (2, 3), (2, 4)},

{(1, 3), (2, 3), (2, 4)}, {(1, 3), (1, 4), (2, 3), (2, 4)}

**9. Let A and B be two sets such that **** and **** If **** are in A **** B.**

**Ans. **Here

A and B

A and B

A and B

But it is given that and

A = and B = {1, 2}

**10. The Cartesian Product A **** A has 9 elements among which are found **** and (0, 1). Find the set A and the remaining elements of A **** A.**

**Ans. **Here

A and A

A and A

A

But it is given that which implies that

A =

And A A =

Therefore, the remaining elements of A A are

and

Nice

Nice