relations and functions class 11

S.No.	RELATIONS AND FUNCTIONS
	EXERCISE 2.1
1	If (x/3 +1 , y -2/3) = (5/3 , 1/3), find the values of x and y.
2	If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A×B).
3	If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
4	State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (i) If P = {m, n} and Q = { n, m}, then P × Q = {(m, n),(n, m)}. (ii) 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. (iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.
5	If A = {–1, 1}, find A × A × A.
6	If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B
7	Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that (i) A × (B ∩ C) = (A × B) ∩ (A × C). (ii) A × C is a subset of B × D.
8	Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.
9	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.
10	The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0,1). Find the set A and the remaining elements of A × A.
..	EXERCISE 2.2
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.
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.
3	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.
4	The Fig2.7 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?
5	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}. (i) Write R in roster form (ii) Find the domain of R (iii) Find the range of R.
6	Determine the domain and range of the relation R defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.
7	Write the relation R = {(x, x3) : x is a prime number less than 10} in roster form.
8	Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.
9	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.
..	EXERCISE 2.3
1	Which of the following relations are functions? Give reasons. 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)}.
2	Find the domain and range of the following real functions: (i) f(x) = –| x| (ii) f(x) = root (9 − x2).
3	A function f is defined by f(x) = 2x –5. Write down the values of (i) f (0), (ii) f (7), (iii) f (–3).
4	The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = 9/5C + 32. Find (i) t(0) (ii) t(28) (iii) t(–10) (iv) The value of C, when t(C) = 212.
5	Find the range of each of the following functions. (i) f (x) = 2 – 3x, x ∈ R, x > 0. (ii) f (x) = x2 + 2, x is a real number. (iii) f (x) = x, x is a real number.
..	Miscellaneous Exercise on Chapter 2
1	The relation f is defined by The relation g is defined by Show that f is a function and g is not a function. (solution of Question 1 )
2	If f (x) = x2, find f ......
3	Find the domain of the function f (x) ....
4	Find the domain and the range of the real function f defined by f (x) = root (x-1)
5	Find the domain and the range of the real function f defined by f (x) = |x –1| .
6	let f .... be a function from R into R. Determine the range of f.
7	Let f, g : R→R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and f/g
8	Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.
9	Let R be a relation from N to 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. Justify your answer in each case.
10	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.
11	Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z? Justify your answer.
12	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.