# Question-1

**Find x and y, if (2x, x+y) = (6, 2).**

**Solution:**

2x = 6

∴ x= 3

x + y = 2

3 + y = 2

∴ y = -1

# Question-2

**Find the domain of the following function : f(x) = x**

**Solution:**

The domain of the function f(x) = x is R.

# Question-3

**Let A = {a, b, c} and B = {p, q}. Find**

(i) A × B

(i) A × B

** (ii) B × A **

** (iii) A × A **

** (iv) B × B **

**Solution:**

(i) A × B = {(a,p), (a,q), (b,p), (b,q), (c,p), (c,q)}

(ii) B × A = {(p,a), (q,a), (p,b), (q,b), (p,c), (q,c)}

(iii) A × A = {(a,a), (a,b), (a,c), (b,a), (b,b), (b,c), (c,a), (c,b), (c,c)}

(iv) B × B = {(p,p), (p,q), (q,p), (q,q)}

# Question-4

**Find the domain of the following function : f(x) =**

**Solution:**

x+|x|= 0 for x < 0 or x = 0.

∴ The domain of the function f(x) = is (0, ∞ )

# Question-5

**Let A = {1, 2, 3}, B = {2, 3, 4} and C = {4, 5}. Verify that**

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

(ii) A × (B ∪ C) = (A × B) ∪ (A × C)

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

(ii) A × (B ∪ C) = (A × B) ∪ (A × C)

**Solution:**

(i) L.H.S = A × (B ∩ C) = {1, 2, 3} × {4} = {(1,4), (2,4), (3,4)}

R.H.S = (A × B) ∩ (A × C)

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

{(1,4), (1,5), (2,4), (2,5), (3,4), (3,5)}

= {(1,4), (2,4), (3,4)}∴ A × (B ∩ C) = (A × B) ∩ (A × C)

(ii) L.H.S = A × (B ∪ C)

= {1, 2, 3} × {2, 3, 4, 5}

= {(1,2), (1, 3), (1, 4), (1,5), (2, 2), (2,3), (2,4), (2,5), (3,2), (3,3),

(3,4), (3,5)}

R.H.S = (A × B) ∪ (A × C)

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

{(1,4), (1,5), (2,4), (2,5), (3,4), (3,5)}

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

∴ A × (B ∪ C) = (A × B) ∪ (A × C)

# Question-6

**If R is the relation "less than" from A = {1, 2, 3, 4, 5} to B = {1, 4, 5}, write down the set of ordered pairs corresponding to R. Find the inverse relation to R.**

**Solution:**

R = {( x, y ) / x ∈A, y ∈B and x < y}

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

.

^{.}. Inverse relation corresponds to the Cartesian product {(4,1),(5, 1),(4,2), (5,2), (4,3), (5,3), (5,4)} and corresponds to the relation 'greater than' from B to A.

# Question-7

**Prove that A ∩ (B - C) = (A ∩ B) - (A ∩ C) .**

**Solution:**

Let x ∈ A ∩ (B - C)

⇔ x ∈ A and x ∈ (B - C)

⇔ x ∈ A and {x ∈ B and x∉ C}

⇔ x ∈ A and {x ∈B and x ∉ C}

⇔ x ∈A and x ∈ B or x ∈ A and x ∉C

⇔( A ∩B) - (A ∩C)

# Question-8

**If A= {1, 2, 3} , B = {4}, C = {5}, then verify that**

(i) A×(B ∪ C) = (A×B) ∪ (A×C)

(ii) A×(B - C) = (A×B) - (A×C)

(i) A×(B ∪ C) = (A×B) ∪ (A×C)

(ii) A×(B - C) = (A×B) - (A×C)

**Solution:**

(i) A×(B ∪ C) = (A×B) ∪ (A×C).

A

*×*(B ∪ C) = {1, 2, 3}

*×*{4, 5}

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

(A

*×*B) ∪ (A

*×*C) = {(1, 4), (2, 4), (3, 4)} ∪ {(1, 5), (2,5), (3, 5)}

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

.

^{.}. A

*×*(B ∪ C) = (A

*×*B) ∪ (A

*×*C)

(ii) A

*×*(B - C) = (A

*×*B) - (A

*×*C)

A

*×*(B - C) = {1, 2, 3}

*×*{4}

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

(A

*×*B) - (A

*×*C) = {(1,4), (2,4), (3,4)} - {(1, 5), (2, 5), (3, 5)}

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

# Question-9

**If R is the relation in N x N defined by (a,b) R (c,d) if and only if a + d = b + c, show that R is an equivalence relation.**

**Solution:**

__Reflexive__

(a,b) R (a,b) ⇔a + b = b + a for a,b ∈ N

⇔ b + a = a + b (Transposing )

⇔ (a,b) R (a,b) for a,b ∈ N

.

^{.}. (a,b) R (a,b) ⇔ (a,b) R (a,b) for a,b ∈ N

__Symmetric__

If (a,b) R (c,d) ⇔a + d = b + c for a,b,c,d ∈N

⇔ b + c = a + d (transposing)

⇔ (c,d) R (a,b) for a,b,c,d∈N

.

^{.}. (a,b) R (c,d) ⇔(c,d) R (a,b) for a,b,c,d ∈N

__Transitive__

If (a,b) R (c,d)⇔ a + d = b + c for a,b,c,d ∈N

and (c,d) R (e,f) ⇔ c + f = d + e for c,d,e,f ∈N

then (a,b) R (c,d) ⇔ a + d = b + c for a,b,c,d ∈N

⇔ a + d + e + f = b + c + e + f

⇔ a + (d + e) + f = b + c + e + f (since c + f = d + e )

⇔ a + f = b + e

⇔ (a,b) R (e,f) for a,b,e,f ∈N

.

^{.}. (a,b) R (c,d) ⇔ (a,b) R (e,f) for a,b,c,d,e,f ∈N

.

^{.}. the relation defined by (a,b) R (c,d) if and only if a + d = b + c is an equivalence relation.

# Question-10

**Find the domain of the following function : f(x) =**

**Solution:**

x

^{2}-3x+2 =0 for x= 2,1.∴ The domain of the function : f(x) = is R- {1,2}.

# Question-11

**Let A = {1, 2, 3, 4} and S = {(a, b): a ∈ A, b ∈ A, a divides b}. Write S explicitly.**

**Solution:**

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

# Question-12

**Find the domain of the following function : f(x) = e**

^{x+sinx}**Solution:**

The domain of the function : f(x) = e

^{x+sinx}is R .

# Question-13

**Find the domain of the following function : f(x) =**

**Solution:**

# Question-14

**Let A = {1, 2} and B = {3, 4}. Write all subsets of A × B.**

**Solution:**

A × B

**=**{(1,3), (1,4), (2,3), (2,4)}.

∴ The subsets of A × B are φ , {(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)}.

# Question-15

**Find the domain of the following function : f(x) = [x] + x**

**Solution:**

The domain of the function f(x) = [x] + x is R.

# Question-16

**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 × B, find A and B, where x, y, z are distinct elements.**

**Solution:**

A = {x, y, z} and B = {1, 2}

# Question-17

**Find the domain of the following function : f(x) =**

**Solution:**

The domain of the function : f(x) = is [−1,1]−{0}

# Question-18

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

**Solution:**

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)}

∴ A × C ⊂ B × D.

# Question-19

**Find the range of each of the following function: f(x) =**

**Solution:**

f(x) =|x-3| is positive for all values of x in R.

The range of the function f(x) = is(0, ∞ ).

# Question-20

**Let A be a non empty set such that A × B = A × C. show that B = C.**

**Solution:**

Let a ∈ A. Since B ≠ φ , there exists b ∈ B. Now, (a, b) ∈ A × B = A × C implies b ∈ C.

∴ every element in B is in C giving B ⊂ C. Similarly, C ⊂ B. Hence B = C.

# Question-21

**Find the range of the following function: f(x) = 1-**

**Solution:**

≥ 0 ⇒ 1- ≤ 1

The range of the function f(x) = 1-is (-∞,1)

# Question-22

**Find the range of the following function: f(x) =**

**Solution:**

f(x) = = 1, if x − 4>0

= −1, if x − 4< 0

∴ The range of the function f(x) = is (−1,1)

# Question-23

**Let A = {1, 2, 3, 4} and B = {x, y, z}. Let R be a relation from A to B defined by R = {(1, x), (1, z), (3, x), (4, y)}. Find the domain and range of R.**

**Solution:**

Domain of R = {1, 3, 4} and Range R = {x, y, z} = B

# Question-24

**Let A = {1, 2, 3, 4} and B = {x, y, z}.
Let R be a relation from A to B defined by R = {(1, x), (1, z), (3, x), (4, y)}.
Draw the arrow diagram of relation R.**

**Solution:**

# Question-25

**Find the range of the following function: f(x) =**

**Solution:**

The range of the function : f(x) = is [0,4] .

# Question-26

**Find the range of the following function: f(x)=**

**Solution:**

The range of the function is (0, ∞)

# Question-27

**In N×N, show that the relation defined by (a,b)R(c,d) if and only if ad = bc is an equivalence relation.**

**Solution:**

__Reflexive__

(a,b)R(a,b) Û ab = ba for a,bÎN

Û ba = ab (Transposing)

Û (a,b)R(a,b) for a,bÎN

.

^{.}. (a,b)R(a,b) ⇔ (a,b)R(a,b) for a,bÎN.

__Symmetric__

(a,b)R(c,d) Û ad = bc a,b,c,dÎN

Û bc = ad (Transposing)

Û (c,d)R(a,b) for a,b,c,dÎN

.

^{.}. (a,b)R(c,d) ⇔ (c,d)R(a,b) for a,b,c,dÎN

__Transitive__

If (a,b)R(c,d) ⇔ ad = bc a,b,c,dÎN

and (c,d)R(e,f) ⇔ c = de c,d,e,fÎN

Then (a,b)R(c,d) ⇔ ad = bc a,b,c,dÎN

Û adef = bcef (Multiplying both sides by ef)

Û adef = be(cf)

Û adef = bede ( Since cf = de)

Û af = be

Û (a,b) R (e,f) a,b,e,fÎN

.

^{.}. (a,b)R(c,d) Û (a,b) R (e,f) a,b,e,fÎN

.

^{.}. the relation defined by (a,b)R(c,d) if and only if ad = bc is an equivalence relation.

# Question-28

**Find the domain and the range of the following function : f(x) =**

**Solution:**

We know that 0 ≤ *x* − [*x*] ≤ 1 for all *x* ∈ R. Also, *x* - [*x*] = 0 for *x* ∈ Z.

is defined if *x* - [*x*] > 0

i.e., *x* ∈ R – Z.

Hence the domain of the function is R – Z.

# Question-29

**Find the domain and the range of the following function : f(x)=**

**Solution:**

-1 ≤sin x ≤1 ⇒ -3 £ 3 sin x ≤ 3

i.e −1 ≤ 4 + 3 sin x ≤ 7

∴

The domain of the function is R; Range :

# Question-30

**Let R be the relation on Z defined by a R b if and only if a – b is an even integer. Find (i) R, (ii) domain R, (iii) range of R.**

**Solution:**

(i) R = {(a, b): a and b are even integers} ∪ {(c, d) : c and d are odd integers}

(ii) Domain = Z

(iii) Range = Z

# Question-31

**Find the domain and the range of the following function : f(x) = 1-**

**Solution:**

The domain of the function is R ; Range : (- ∝ ,1)

# Question-32

**Let R be the relation on Z defined by R = {(a, b): a ∈ Z, b ∈ Z, a**

^{2}= b^{2}}. Find (i) R, (ii) domain R, (iii) range of R.**Solution:**

(i) R = {(a, a): a ∈ Z} ∪ {(a, -a): a ∈ Z}

(ii) Domain = Z

(iii) Range = Z

# Question-33

**Find the domain and the range of the following function : f(x) = x!**

**Solution:**

The domain of the function is N ∪ {0}; Range : {n! : n = 0,1,2…..}

# Question-34

**Determine the domain and the range of the relation R defined by R = {(x+1, x+5): x ∈ {0,1 ,2, 3, 4,5}}**

**Solution:**

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

# Question-35

**Determine the domain and the range of the relation R, where R = {(x, x**

^{3}) : x is a prime number less than 10}.**Solution:**

Domain = {2, 3, 5, 7}, Range = {8, 27, 125, 343}

# Question-36

**Find the domain and the range of the following function : f(x) = sin**

^{2}(x^{3}) + cos^{2}(x^{3})**Solution:**

sin

^{2}(x

^{3}) + cos

^{2}(x

^{3}) =1

∴ The domain of the function is R ; Range : {1}

# Question-37

**Is inclusion of a subset in another, i.e., ARB if and only if A ⊂ B, in the context of a universal set, an equivalence relation in the class of subsets of the universal set? Justify your answer.**

**Solution:**

Let U be the universal set .Let R be the relation ' is a subset of' or 'is included in ' between the subsets of U.

Since every set is a subset of itself i.e., for every subset A in U, A ⊆ A or A R A.

.

^{.}. R is reflexive.

Now let A and B be two subsets of U such that A ⊆ B, then it is not necessary that B must also be a subset of A.

.

^{.}. A R B need not imply B R A.

.

^{.}. R is not symmetric.

Hence R is not an equivalence relation.

# Question-38

**Find the domain and the range of the following function : f(x) =**

**Solution:**

f(x) = = x+3

The domain of the function is R ; Range : R .

# Question-39

**Determine the domain and range of the following relations**

** (i) {(1, 2), (1, 4), (1, 6), (1, 8)} **

** (ii) {(x, y) : x ∈ N, y ∈ N and x + y = 10}**

** (iii) {(x, y) : x ∈ N, x<5, y = 3}**

** (iv) {(x, y) : y = |x – 1|, x ∈ Z and |x| ≤ 3}**

**Solution:**

(i) Domain = {1}, Range = {2, 4, 6, 8}

(ii) Domain = {1, 2, 3, 4, 5, 6, 7, 8, 9}, Range = {9, 8, 7, 6, 5 ,4, 3, 2, 1}

(iii) Domain = {1, 2, 3, 4}, Range = {3}

(iv) Domain = {-3, -2, -1, 0, 1, 2, 3}, Range = {4, 3, 2, 1, 0}

# Question-40

**How many relations are possible from a set A of m elements to another set B of n elements? Why?**

**Solution:**

Number of elements in A = m.

Number of elements in B = n

.

^{.}. Number of elements in A×B = mn

Number of subsets of A×B = 2

^{mn}

Since every subset of A×B is a relation from A to B therefore 2

^{mn}relations are possible from A to B.

# Question-41

**Draw the graph of the following function: f(x) = ,x ≠ 0**

**Solution:**

# Question-42

**Let A = {1, 2}. List all the relations on A.**

**Solution:**

The relations on A are

# Question-43

**Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A into B.**

**Solution:**

n (A) = 3 and n (B) = 2

∴n (A × B) = 2 × 3 = 6

∴the number of relations from A into B are 2

^{6}= 64.

# Question-44

**Which of the following relations are functions? If it is a function, determine its domain and range:**

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

** (ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}**

** (iii) {(0, 0), (1, 1), (1, -1), (4, 2), (4, -2), (9, 3), (9, -3), (16, 4), (16, -4)}**

** (iv) {(1, 2), (1, 3), (2, 5)}**

** (v) {(2, 1), (3, 1), (5, 2)}**

** (vi) {(1, 2), (2, 2), (3, 2)}**

**Solution:**

(i) Domain = {2, 5, 8, 11, 14, 17}, Range = {1}

(ii) Domain = {2, 4, 6, 8, 10, 12, 14}, Range = {1, 2, 3, 4, 5, 6, 7}

(iii) No, As there are four pairs of ordered pairs which have the same first element.

(iv) No, As two ordered pairs which have the same first element.

(v) Domain = {2, 3, 5}, Range = {1, 2}

(vi) Domain = {1, 2, 3}, Range = {2}

# Question-45

**If A = {1,2,3}, B = {a, b}, find A × A .**

**Solution:**

A × A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

# Question-46

**Find the domain and range of the following functions:**

** (i) **

** (ii) {(x, -|x|): x ∈ R} **

**Solution:**

(i) Domain = R – {1}, Range = R – {2}

(ii) Domain = R, Range = {y: y ∈ R and y ≤ 0}

# Question-47

**If A = {1,2,3}, B = {a, b}, find A × B**

**Solution:**

A × B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}

# Question-48

**Find the domain and range of the following functions:**

** (i) {(x, ): x∈ R }**

** (ii) **

**Solution:**

(i) Domain = {x: x ∈ R and –3 ≤ x ≤ 3}, Range = {y: y ∈ R and –3 ≤ y ≤ 3}

(ii) Domain = R – {1, -1}, Range = {y: y ∈ R, y ≠ 0, y<0 and y ≥ 1}

# Question-49

**If A = {1,2,3}, B = {a, b}, find B × B.**

**Solution:**

B × B = {(a, a), (a, b), (b, a), (b, b)}

# Question-50

**If f(x) = x**

(i) find f(2) × f(-2) (ii) find (iii) find x so that f(x) = 8

^{2}– 1,(i) find f(2) × f(-2) (ii) find (iii) find x so that f(x) = 8

**Solution:**

(i) f(2) × f(−2) = (2

^{2}− 1) [(−2)

^{2}− 1)] = (4 −1) (4 −1) = 3× 3 =9

(ii)

(iii) f(x) = 8

∴x

^{2}– 1 = 8

x

^{2}= 8 + 1

x

^{2}= 9

x = ± 3

# Question-51

**Find f + g, f – g, α f(α ∈ R) ,f.g, and , if f(x)= x**

^{3}+ 1 ; g(x) = x+1**Solution:**

f(x) = x

^{3}+ 1 ; g(x) = x+1

f + g= x^{3} + 1+ x+1 = x^{3} + x + 2

f – g= x^{3} + 1– x – 1 = x^{3} – x

α f(α ∈ R) = ∝ (x^{3} + 1)

f.g = (x^{3} + 1)(x+1) = x^{4} + x^{3} + x + 1

= ,x ≠ –1,

= ≠ –1.

# Question-52

**Find f + g, f – g, α f(α ∈ R) ,f.g, and , if f(x) = cosx ; g(x) = e**

^{x}.**Solution:**

f(x) = cosx ; g(x) = e

^{x}.

f + g= cosx + e^{x }

f – g= cosx - e^{x }

α f(α ∈ R) = α cosx

f.g = e^{x} cosx

= 1/cosx , x ≠ (2n+1)π /2 , n∈ z

= e^{- x} cosx

# Question-53

**Find the range of the following functions: (i) cosx - sinx (ii)**

**Solution:**

(i) f(x) = cosx − sinx

==

=

-1 ≤

∴ −

−

∴Range of the given problem is [−]

(ii) f(x) =

−1 ≥ cos3x ≥ 1

∴4 ≤ 5 – cos3x ≤ 6

Domain =

# Question-54

**Find the domain of +**

**Solution:**

Domain of f(x) = is (−2, ∞ )

g(x) = will be defined for (1-x) > 0 and (1-x) ≠ 1

∴x < 0 and x ≠ 0

Domain of g(x) is (− ,0) ∪ (0,1)

Hence domain of + is (−, 0) ∪ (0,1) ∩[-2,∞) i.e., [-2,0) ∪ (0,1)

# Question-55

**Let R be a relation from N into N defined by R = {(a, b): a, b ∈ N and a = b**

(i) (a, a) ∈ R, for all a ∈ N.

^{2}}. Are the following true(i) (a, a) ∈ R, for all a ∈ N.

** (ii) (a, b) ∈ R implies (b, a) ∈ R.
(iii) (a, b) ∈ R, (b ,c) ∈ R implies (a, c) ∈ R **

**Solution:**

(i)

**a ≠ a**

^{2}. ∴ It is false.

(ii) a = b^{2}. So b ≠ a^{2}. ∴ It is false.

(iii) a = b^{2} and b = c^{2}. Hence a ≠ c^{2∴ }It is false.

# Question-56

**Let f = {(1, 1), (2, 3), (3, 5), (4, 7)} be a function from Z into Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.**

**Solution:**

(1, 1) ∈ f implies f(1) = 1

and (2, 3) ∈ f implies f(2) = 3

So, a + b = 1

and 2a + b = 3

∴ a = 2 and b = -1.

Then, f(3) = 2(3) + (-1) = 6 – 1 = 5, and f(4) = 7.

# Question-57

**Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?**

(i) f is a relation from A to B

(i) f is a relation from A to B

** (ii) f is a function from A to B**

**Justify your answer in each case.**

**Solution:**

(i) True, f is a relation from A into B. Since f is the subset of A ×B.

(ii) No, Since two ordered pairs (2, 9) and (2, 11) in f have the same first component, f is not a function from A to B.

# Question-58

**Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n. Find the range of f.**

**Solution:**

The prime factors of 9 are 3 and 3. ∴f(9) = 3

The prime factors of 10 are 2 and 5. ∴ f(10) = 5

The prime factor of 11 is 11. ∴ f(11) = 11

The prime factors of 12 are 2, 3 and 3. ∴ f(12) = 3

The prime factor of 13 is 13. ∴ f(13) = 13

∴ Range of f = {3, 5, 11, 13}

# Question-59

**Let f: N – {1} → N defined by f(n) = the highest prime factor of n. Find the range of f.**

**Solution:**

Range of f = The set of all prime numbers.

# Question-60

**Let A ⊆ N and f : A → A be defined by f(n) = p, the highest prime factor of n such that the range of f is A. Determine A.**

**Solution:**

A is a set of some prime number.

# Question-61

**Let A = {1, 2, 3}. Find all one – to – one functions from A to A.**

**Solution:**

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

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

# Question-62

**Draw the graph of the function f(x) =**

**Solution:**

# Question-63

**If A= {l, m, n} , B = {x}, C = {y}, then verify that**

(i) A×(B ∪ C) = (A×B) ∪ (A×C)

(ii) A×(B ∩ C) = (A×B) ∩ (A×C)

(iii) A×(B - C ) = (A×B) - (A×C)

(i) A×(B ∪ C) = (A×B) ∪ (A×C)

(ii) A×(B ∩ C) = (A×B) ∩ (A×C)

(iii) A×(B - C ) = (A×B) - (A×C)

**Solution:**