The document NCERT Solution - Relations and Functions (Ex - 2.1) JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 11.

All you need of JEE at this link: JEE

**NCERT QUESTION**

**(Ex - 2.1) **

**Ques 1: ****If ****, find the values of x and y. **

**Ans: **It is given that .

Since the ordered pairs are equal, the corresponding elements will also be equal.

Therefore, and .

âˆ´ *x* = 2 and *y* = 1

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

**Ans: **It is given that set A has 3 elements and the elements of set B are 3, 4, and 5.

â‡’ Number of elements in set B = 3

Number of elements in (A Ã— B)

= (Number of elements in A) Ã— (Number of elements in B)

= 3 Ã— 3 = 9

Thus, the number of elements in (A Ã— B) is 9.

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

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

We know that the Cartesian product P Ã— Q of two non-empty sets P and Q is defined as

P Ã— Q = {(*p*, *q*): *p*âˆˆ P, *q* âˆˆ Q}

âˆ´G Ã— H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

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

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

**(i) If P = { m, n} and Q = {n, m}, then P Ã— Q = {(m, n), (n, m)}.**

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

**(iii) If A = {1, 2}, B = {3, 4}, then A Ã— (B âˆ© Î¦) = Î¦.**

**Ans: **(i) False

If P = {*m*, *n*} and Q = {*n*, *m*}, then

P Ã— Q = {(*m*, *m*), (*m*, *n*), (*n**,* *m*), (*n*, *n*)}

(ii) True

(iii) True

**Ques 5: ****If A = {â€“1, 1}, find A Ã— A Ã— A.**

**Ans: **It is known that for any non-empty set A, A Ã— A Ã— A is defined as

A Ã— A Ã— A = {(*a*, *b*, *c*): *a*, *b*, *c* âˆˆ A}

It is given that A = {â€“1, 1}

âˆ´ A Ã— A Ã— A = {(â€“1, â€“1, â€“1), (â€“1, â€“1, 1), (â€“1, 1, â€“1), (â€“1, 1, 1),

(1, â€“1, â€“1), (1, â€“1, 1), (1, 1, â€“1), (1, 1, 1)}

**Ques 6: ****If A Ã— B = {( a, x), (a, y), (b, x), (b, y)}. Find A and B.**

**Ans: **It is given that A Ã— B = {(*a*, *x*), (*a,* *y*), (*b*, *x*), (*b*, *y*)}

We know that the Cartesian product of two non-empty sets P and Q is defined as P Ã— Q = {(*p*, *q*): *p* âˆˆ P, *q* âˆˆ Q}

âˆ´ A is the set of all first elements and B is the set of all second elements.

Thus, A = {*a*, *b*} and B = {*x*, *y*}

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

**(i) A Ã— (B âˆ© C) = (A Ã— B) âˆ© (A Ã— C)**

**(ii) A Ã— C is a subset of B Ã— D**

**Ans: **(i) To verify: A Ã— (B âˆ© C) = (A Ã— B) âˆ© (A Ã— C)

We have B âˆ© C = {1, 2, 3, 4} âˆ© {5, 6} = Î¦

âˆ´L.H.S. = A Ã— (B âˆ© C) = A Ã— Î¦ = Î¦

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

A Ã— C = {(1, 5), (1, 6), (2, 5), (2, 6)}

âˆ´ R.H.S. = (A Ã— B) âˆ© (A Ã— C) = Î¦

âˆ´L.H.S. = R.H.S

Hence, A Ã— (B âˆ© C) = (A Ã— B) âˆ© (A Ã— C)

(ii) To verify: A Ã— C is a subset of B Ã— D

A Ã— C = {(1, 5), (1, 6), (2, 5), (2, 6)}

B Ã— 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)}

We can observe that all the elements of set A Ã— C are the elements of set B Ã— D.

Therefore, A Ã— C is a subset of B Ã— D.

**Ques 8: ****Let A = {1, 2} and B = {3, 4}. Write A Ã— B. How many subsets will A Ã— B have? List them.**

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

âˆ´A Ã— B = {(1, 3), (1, 4), (2, 3), (2, 4)}

â‡’ *n*(A Ã— B) = 4

We know that if C is a set with *n*(C) = *m*, then *n*[P(C)] = 2* ^{m}*.

Therefore, the set A Ã— B has 2^{4} = 16 subsets. These are

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

**Ques 9: ****Let A and B be two sets such that n(A) = 3 and n (B) = 2. If (x, 1), (y, 2), (z, 1) are in A Ã— B, find A and B, where x, y and z are distinct elements.**

**Ans: **It is given that *n*(A) = 3 and *n*(B) = 2; and (*x*, 1), (*y*, 2), (*z*, 1) are in A Ã— B.

We know that A = Set of first elements of the ordered pair elements of A Ã— B

B = Set of second elements of the ordered pair elements of A Ã— B.

âˆ´ *x*, *y*, and *z* are the elements of A; and 1 and 2 are the elements of B.

Since *n*(A) = 3 and *n*(B) = 2, it is clear that A = {*x*, *y*, *z*} and B = {1, 2}.

**Ques 10: ****The Cartesian product A Ã— A has 9 elements among which are found (â€“1, 0) and (0, 1). Find the set A and the remaining elements of A Ã— A.**

**Ans: **We know that if *n*(A) = *p* and *n*(B) = *q,* then *n*(A Ã— B) = *pq*.

âˆ´ *n*(A Ã— A) = *n*(A) Ã— *n*(A)

It is given that *n*(A Ã— A) = 9

âˆ´ *n*(A) Ã— *n*(A) = 9

â‡’ *n*(A) = 3

The ordered pairs (â€“1, 0) and (0, 1) are two of the nine elements of A Ã— A.

We know that A Ã— A = {(*a, a*): *a* âˆˆ A}. Therefore, â€“1, 0, and 1 are elements of A.

Since *n*(A) = 3, it is clear that A = {â€“1, 0, 1}.

The remaining elements of set A Ã— A are (â€“1, â€“1), (â€“1, 1), (0, â€“1), (0, 0),

(1, â€“1), (1, 0), and (1, 1)

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!