NCERT solutions Mathematics Class 11 chapter 2 Relations and Functions

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

if `(x/3 + 1, y - 2/3) = (5/3, 1/3)` find the values of x and y.

State whether the following statement are true or false. If the statement is false, rewrite the given statement correctly.

If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}.

State whether the following statement are true or false. If the statement is false, rewrite the given statement correctly.

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.

State whether the following statement are true or false. If the statement is false, rewrite the given statement correctly.

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 A × (B ∩ C) = (A × B) ∩ (A × C)

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that   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 given figure shows a relationship between the sets P and Q. write this relation

(i) in set-builder form (ii) in 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.

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.

(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)}

Find the domain and range of the given real function:

f(x) = –|x|

Find the domain and range of the following real function:

f(x) = `sqrt(9 - x^2)`

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 `t(C) = "9C"/5 + 32`

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.

Find the range of each of the following functions.

f(x) = x, x is a real number

Find the range of each of the following functions.

f(x) = x² + 2, x, is a real number.

The relation f is defined by f(x) = `{(x^2,0<=x<=3),(3x,3<=x<=10):}`

The relation g is defined by  g(x) = `{(x^2, 0 <= x <= 2),(3x,2<= x <= 10):}`

If f(x) = x², find `(f(1.1) - f(1))/((1.1 - 1))`

Find the domain of the function  f(x) = `(x^2 + 2x + 1)/(x^2 - 8x + 12)`

Find the domain and the range of the real function f defined by `f(x)=sqrt((x-1))`

Let `f = {(x, x^2/(1+x^2)):x ∈ R}` 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 `f/g`

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 ∈ 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.

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.

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.