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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

        (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)

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)

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,dN

... (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:

x2-3x+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) = ex+sinx

Solution:

The domain of the function : f(x) = ex+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, a2 = b2}. 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, x3) : 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) = sin2(x3) + cos2(x3)

Solution:
sin2(x3) + cos2(x3) =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 = 2mn

Since every subset of A×B is a relation from A to B therefore 2mn 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 26 = 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)}

          (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) = x2 – 1,   

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

Solution:
(i) f(2) × f(2) = (22 1) [(2)2 1)] = (4 1) (4 1) = 3× 3 =9

(ii)

(iii) f(x) = 8


x2 – 1 = 8

x2 = 8 + 1

x2 = 9

x = ± 3

Question-51

Find f + g, f – g, α f(α R) ,f.g, and , if f(x)= x3 + 1 ; g(x) = x+1

Solution:
f(x) = x3 + 1 ; g(x) = x+1

f + g= x3 + 1+ x+1 = x3 + x + 2

f – g= x3 + 1– x – 1 = x3 – x

α f(α R) = (x3 + 1)

f.g = (x3 + 1)(x+1) = x4 + x3 + 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) = ex.

Solution:
f(x) = cosx ; g(x) = ex.

f + g= cosx + ex

f – g= cosx - ex

α f(α R) = α cosx

f.g = ex 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 = b2}. 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 a2. It is false.

(ii) a = b2. So b a2. It is false.

(iii) a
= b2 and b = c2. Hence a c2 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

         (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)

Solution:





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