If , find the values of x and y.
If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A×B).
If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
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.
If A = {–1, 1}, find A × A × A.
If A × B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B.
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.
(ii) A × C is a subset of B × D.
Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.
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.
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.
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.
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.
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.
The Fig 2.7 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?
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.
Determine the domain and range of the relation R defined by
R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.
Write the relation R = {(x, x3) : x is a prime number less than 10} in roster form.
Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.
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.
Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
{(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}
{(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)}
{(1,3), (1,5), (2,5)}.
Find the domain and range of the following real functions:
(i) f(x) = -|x|
A function f is defined by f (x) = 2x –5. Write down the values of
(i) f (0), (ii) f (7), (iii) f (–3).
The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by
Find (i) t(0) (ii) t(28) (iii) t(–10) (iv) The value of C, when t(C) = 212.
Find the range of each of the following functions.
f(x) = 2 – 3x, x ∈ R, x > 0.
f(x) = x2 + 2, x is a real number.
f(x) = x, x is a real number.
The relation f is defined by
The relation g is defined by
Show that f is a function and g is not a function.
If f (x) = x2, find
Find the domain of the function
Find the domain and the range of the real function f defined by .
Find the domain and the range of the real function f defined by.
Let be a function from R into R. Determine the range of f.
Let f, g : R → R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f - g and .
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.
Let R be a relation from N to N defined by R = {(a, b): a, b, εN and a = b2}. Are the following true?
(i) (a,a) ε R, for all a ε N
(ii) (a,b) ε R, implies (b,a) ε R
(iii) (a,b) ε R, (b,c) ε R implies (a,c) ε R.
Justify your answer in each case.
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.
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.
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.
