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Question-1

If , Find the values of x and y.

Solution:
+1 =
=

x = 5 - 3
x = 2
y - =
y = +
y = = 1
x = 2, y =1.

Question-2

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

Solution:
(AB) will have 33 = 9 elements.

Question-3

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

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

Question-4

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.
a. If P = {m,n} and Q = {n,m}, then P Q = {(m,n),(n,m)}.
b. If A and B are non-empty sets, then A B is a non- empty set of ordered pairs (x,y) such that x A and y B.
c. If A = {1,2), B = {3,4}, then A (B ) = .

Solution:
a. False.
P Q = { (m n) (mm) (n n) (n m)}
b. True.

c. True.

Question-5

If A = {-1,1}, Find A A A.

Solution:
A A = {-1,1} {-1,1}
= {(-1,-1) (-1,1) (1,-1) (1,1)}
A A A = {(-1,-1) (-1,1) (1,-1) (1,1)} {-1,1}
=

Question-6

If A B = {(a,x),(a,y),(b,x),(b,y)}, Find A and B.

Solution:
A = {a,b} B={x,y}

Question-7

Let A = {1, 2} and B = {3, 4}. Write A X B.  How many subsets will A x B have ?

Solution:
A × B = {(1,3), (1,4), (2,3), (2,4)}.

A × B will have 24 = 16 subsets.

Question-8

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 and z are distinct elements.

Solution:
A = {x,y,z} B = {1,2}

Question-9

The Cartesian product A x A has 9 elements among which are found (-1,0) and (0, 1). Find the set A and the remaining elements of A x A.

Solution:
A = {-1, 0, 1};
The remaining elemnts of A x A are = {-1, 0}; (-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0), (1, 1).

Question-10

Let A = {1,2,3….14}. Define a relation R from A to A by R = {(x,y) : 3x - y = 0, where x,yA}. Write down its domain, codomain and range.

Solution:
3x-y = 0
 

x

1

2

3

4

y

3

6

9

12

R = {(1,3) (2,6) (3,9) (4,12)}
Domain of R = {1,2,3,4}
Range of R = {3,6,9,12}
Codomain of R = {1,2,3…,14}

Question-11

Define a relation R on the set N of natural numbers by R = {(x,y):y= x + 5, x is a natural number less than 4: x,y N}. Depict this
relationship using roster form. Write down the domain and the range.

Solution:
R = {(1,6) (2,7) (3,8)}
Domain of R = {1,2,3}
Range of R = {6,7,8}

Question-12

A = {1,2,3,5} and B = { 4, 6, 9}. Define a relation R from A to B by R= {(x,y): the difference between x and y is odd: x A, y B }.Write R in roster form.

Solution:
R = { (1,4),(1,6), (2,9), (3,4), (3,6), (5,4), (5,6)}

Question-13

What is its domain and range?

 


Solution:
(i) Set-builder form:R = {(x,y); y = x - 2 for x= 5,6,7 }
(ii) Roster form: R = {(5,3), (6,4), (7,5)}
Domain = {5,6,7}
Range = {3,4,5}

Question-14

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by{(a, b) : a, b A, b is exactly divisible by a}.  Find (i) R in roster form, (ii) domain of R, (iii) range of R.

Solution:
(i) R = {(1,1), (1, 2), (1,3), (1,4), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (6,6)}
(ii) Domain = A
(iii) Range = A

Question-15

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

Solution:
R = {(0,5), (1,6), (2,7), (3,8) (4,9), (5,10)}
Domain of R = { 0,1,2,3,4,5}
Range of R = {5,6,7,8,9,10}

Question-16

Write the relation R, where R = {(x,x3) : x is a prime number less than 10} in roster form.

Solution:
R={(2, 8), (3, 27), (5, 125), (7, 343)}

Question-17

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 x B) = 3 x 2 = 6
The number of relations from A to B = 26 = 64.

Question-18

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-19

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

Solution:

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

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

(iii) No, as two ordered pairs in the relation have the same first element.

Question-20

Find the domain and range of the following real functions:
(i) f(x) = -lxl (ii) f(x) = .

Solution:
(i) Domain = {R}
Range = {-
}
(ii) f(x) =
Domain of function = {x : -3
x 3}
Range of function = {x : 0
x 3}

Question-21

A function f is defined by f(x) = 2x - 5. Write down the values of
(i) f(0), (ii) f(7), (iii) f(–3).

Solution:
f(x) = 2x – 5
(i) f(0)= 2(0) – 5
         = 0 – 5
         = –5

(ii) f(7)= 2(7) – 5
          = 14 – 5
          = 9

(iii) f(–3)= 2(–3) – 5
            =–6–5
            = –11

Question-22

The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = + 32.
Find
(i) t(0)
(ii) t(28)
(iii) t(
10)
(iv) The value of C, when t(C) =212.

Solution:
t(C) = + 32
(i) t(0) = + 32
          = 0 + 32 = 32
(ii) t(28) = + 32
            = + 32
            = 50.4 + 32
            = 82.4
(iii) t(
10) = + 32
               = -18 + 32
               = 14.
(iv) The value of C, when t(C) =212
212 = + 32
212 5 = 9C + 160
1060 – 160 = 9C
          900 = 9C
                 C= 100




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