Referring to the AP Inter 1st Year Maths Study Material Chapter 2 Relations and Functions Exercise 2b Solutions makes it easier to understand complex problems.
AP Inter 1st Year Maths Relations and Functions Solutions Exercise 2b
I.
Question 1.
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 e A, y ∈ B}. Write R in roster form.
Solution:
Given A = {1, 2, 3, 5}, B= {4, 6, 9}
R = {(x, y): the difference between x & y is odd; x ∈ A, y ∈ B}.
Roster form of R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}.
Question 2.
Determine the domain and range of the relation R defined by R = {(x, x + 5): x ∈ {0, 1, 2, 3, 4, 5}}.
Solution:
Given R = {(x, x + 5): x ∈ {0, 1, 2, 3, 4, 5}}
The roster form of 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 3.
Write the relation R = {(x, x3) : x is a prime number less than 10} in roster form.
Solution:
Given R = {(x, x3) ; x is a’prime number less than 10}
The prime numbers less than 10 are 2, 3, 5 & 7
∴ The roster form R = {(2, 8), (3, 27), (5, 125), (7, 343)}
Question 4.
Let A = (x, y, z) and B = (1, 2). Find the number of relations from A to B.
Solution:
Given A = {x, y, z} & B = {1, 2}
A × B = {(x, 1), (x, 2), (y, 1), (y, 2), (z, 1), (z, 2)}
n (A × B) = 6, n(B) = 2, the number of subsets of A × B is 26.
The number of relations from A to B is 26.
Question 5.
Let R be the relation on Z defined by R = ((a,b): a, b ∈ Z, a – b is an integer). Find the domain and range of R.
Solution:
Given R = {(a, b): a, b ∈ Z, a – b is an integer}
∴ Domain of R = Z, Range of R = Z
II.
Question 1.
Let A = {1, 2, 3,…, 14}. Define a relation R from A to A by R = {(x, y) : 3x – y = 0, where x, y ∈ A). Write down its domain, codomain and range,
Solution:
A relation R from A to A is given by R = {(x, y) ; 3x – y = 0, where x, y ∈ A}
The roster form is given by R = {(1, 3), (2, 6), (3, 9), (4, 12)}
The domain of R = {1, 2, 3, 4}
The whole set A is the co-domain of the relation R.
∴ Co-domain of R = {1, 2, 3, 14}
The range of R = {3, 6, 9, 12}.
Question 2.
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:
Given R = {(x, y): y = x + 5, x < 4;x, y ∈ N)
⇒ x = 1, 2, 3 R = {(1, 6), (2, 7), (3, 8)}
The domain of R = {1, 2, 3},
The range of R = {6, 7, 8}
Question 3.
The Fig. shows a relationship between the sets P and Q. Write this relation,
i) in set-builder form ii) roster form. What is its domain and range ?
Solution:
Given P = {5, 6, 7} Q = {3, 4, 5}
i) The set builder form
R = {(x, y): y = x – 2 for x = 5, 6, 7}
ii) The roster form R = {(5, 3), (6, 4), (7, 5)}
Domain of R = {5, 6, 7}
Range of R = {3, 4, 5}
Question 4.
Let A = {1, 2, 3, 4, 6}. Let R be the relation on A is defined by {(a, b): a , b ∈ A, b is exactly divisible by a}.
(i) Write R in roster form
(ii) Find the domain of R
(iii) Find the range of R.
Solution:
Given A = {1, 2, 3, 4, 6} and R = {(a, b) : a, b ∈ A, b is exactly divisible by a}
i) The roster form 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 of R = {1, 2, 3, 4, 6}
iii) Range of R = {1, 2, 3, 4, 6}