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.
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.
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)}
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.
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}
=
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}
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 Ã— B = {(1,3), (1,4), (2,3), (2,4)}.
A Ã— B will have 2^{4} = 16 subsets.
Solution:
A Ã— B = {(1,3), (1,4), (2,3), (2,4)}.
A Ã— B will have 2^{4} = 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}
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).
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 = {1,2,3â€¦.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
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 = {1,2,3â€¦,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}
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)}
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}
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 âˆˆ 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
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}
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,x^{3}) : x is a prime number less than 10} in roster form.
Solution:
R={(2, 8), (3, 27), (5, 125), (7, 343)}
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
âˆ´n (A x B) = 3 x 2 = 6
âˆ´The number of relations from A to B = 2^{6} = 64.
Solution:
n(A) = 3 and n(B) = 2
âˆ´n (A x B) = 3 x 2 = 6
âˆ´The number of relations from A to B = 2^{6} = 64.
Question-18
Let R be the relation on Z defined by a R b if and only if a â€“ 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} âˆª {(c, d): c and d are odd integers}
(ii) Domain = Z
(iii) Range = Z
integer. Find (i) R, (ii) domain R, (iii) range of R.
Solution:
(i) R = {(a, b): a and b are even integers} âˆª {(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:
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 = {-âˆž}
(ii) f(x) =
Domain of function = {x : -3 â‰¤ x â‰¤ 3}
Range of function = {x : 0 â‰¤x â‰¤3}
(i) f(x) = -lxl (ii) f(x) = .
Solution:
(i) Domain = {R}
Range = {-âˆž}
(ii) f(x) =
Domain of function = {x : -3 â‰¤ x â‰¤ 3}
Range of function = {x : 0 â‰¤x â‰¤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(â€“3).
Solution:
f(x) = 2x â€“ 5
(i) f(0)= 2(0) â€“ 5
= 0 â€“ 5
= â€“5
(ii) f(7)= 2(7) â€“ 5
= 14 â€“ 5
= 9
(iii) f(â€“3)= 2(â€“3) â€“ 5
=â€“6â€“5
= â€“11
(i) f(0), (ii) f(7), (iii) f(â€“3).
Solution:
f(x) = 2x â€“ 5
(i) f(0)= 2(0) â€“ 5
= 0 â€“ 5
= â€“5
(ii) f(7)= 2(7) â€“ 5
= 14 â€“ 5
= 9
(iii) f(â€“3)= 2(â€“3) â€“ 5
=â€“6â€“5
= â€“11
Question-22
The function â€˜tâ€™ 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(âˆ’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(âˆ’10) = + 32
= -18 + 32
= 14.
(iv) The value of C, when t(C) =212
212 = + 32
212 5 = 9C + 160
1060 â€“ 160 = 9C
900 = 9C
C= 100
Find
(i) t(0)
(ii) t(28)
(iii) t(âˆ’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(âˆ’10) = + 32
= -18 + 32
= 14.
(iv) The value of C, when t(C) =212
212 = + 32
212 5 = 9C + 160
1060 â€“ 160 = 9C
900 = 9C
C= 100