CBSE Class 11 Applied Mathematics Relations and Functions Notes Set 01

Cartesian Product of Sets

Ordered Pair

Two numbers a and b listed in a specific order and enclosed in parentheses form an ordered pair (a, b). In the ordered pair (a, b), we call a the first number and b the second number.
Thus (a, b) ≠ (b, a).
Equality of two ordered pairs: We have (a, b) = (c, d) ⇔ a = c and b = d.

Cartesian Product of Two Sets

Let A and B be two non-empty sets. Then the Cartesian product of A and B is the set, denoted by (A × B), consisting of all ordered pairs (a, b) such that a ∈ A and b ∈ B.
∴ (A × B) = {(a, b) : a ∈ A and b ∈ B}
If A = ∅ or B = ∅, we define A × B = ∅.
Remarks:
(i) If n(A) = p and n(B) = q, then n(A × B) = pq and n(B × A) = pq.
(ii) If at least one of A and B is infinite, then (A × B) is infinite and (B × A) is infinite.

Example 1. If (x + 1, y − 2) = (3, 1), find the values of x and y.
Answer:
Since the ordered pairs are equal, the corresponding elements are equal. Therefore x + 1 = 3 and y − 2 = 1.
Solving, we get x = 2 and y = 3.

Example 2. If A × B = {(p, q), (p, r), (m, q), (m, r)}, find A and B.
Answer:
A = set of first elements = {p, m}
B = set of second elements = {q, r}

Ordered Triplet

Three numbers a, b and c listed in a specific order and enclosed in parentheses form an ordered triplet (a, b, c).
Thus (1, 2, 3) ≠ (2, 1, 3) ≠ (3, 2, 1) etc.
For any non-empty set A, we define (A × A × A) = {(a, b, c) : a, b, c ∈ A}.

Example 3. If P = {1, 2}, form the set P × P × P.
Answer:
P × P = {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}
P × P × P = (P × P) × P = {(1, 1), (1, 2), (2, 1), (2, 2)} × {1, 2}
= {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}

Example 4. If R is the set of all real numbers, what do the Cartesian products R × R and R × R × R represent?
Answer:
The Cartesian product R × R = {(x, y) : x, y ∈ R} represents the coordinates of all the points in two-dimensional space.
The Cartesian product R × R × R = {(x, y, z) : x, y, z ∈ R} represents the coordinates of all the points in three-dimensional space.

Relations

Relation

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B.
Thus R is a relation from A to B ⇔ R ⊆ (A × B). If (a, b) ∈ R, then we say that 'a' is related to 'b' and we write a R b.

Domain, Range and Co-domain of a Relation

Let R be a relation from A to B. Then R ⊆ (A × B).
(i) The set of all first coordinates of elements of R is called the domain of R, written as dom (R).
(ii) The set of all second coordinates of elements of R is called the range of R, denoted by range (R).
(iii) The set B is called the co-domain of R.
dom (R) = {a : (a, b) ∈ R} and range (R) = {b : (a, b) ∈ R}

Total Number of Relations from A to B

Let n(A) = p and n(B) = q. Then n(A × B) = pq. We know that every subset of A × B is a relation from A to B.
∴ Total number of subsets of A × B = 2^pq
∴ Total number of relations from A to B = 2^pq

Representation of a Relation

Let A and B be two given sets. Then a relation R ⊆ A × B can be represented in any of the forms given below.
(1) Roster form: In this form R is given as a set of ordered pairs.

Example 5. Let A = {−2, −1, 0, 1, 2}, B = {0, 1, 4, 9} and R = {(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)}.
(i) Show that R is a relation from A to B.
(ii) Find dom (R), range (R) and co-domain of R.
Answer:
(i) Since R ⊆ A × B, R is a relation from A to B. Note that −2 R 4, −1 R 1, 0 R 0, 1 R 1 and 2 R 4.
(ii) dom (R) = set of first coordinates of elements of R = {−2, −1, 0, 1, 2}
range (R) = set of second coordinates of elements of R = {0, 1, 4}
Co-domain of R = {0, 1, 4, 9} = B

(2) Set-builder form: Under this method, for every (a, b) ∈ R, a general relation is given between a and b. Using this relation, all the elements of R can be obtained.

Example 6. Let A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation from A to B given by R = {(a, b) : a ∈ A, b ∈ B and (a − b) is odd}.
(i) Write R in roster form.
(ii) Find dom (R) and range (R).
Answer:
(i) Clearly, we have R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}
(ii) dom (R) = set of first coordinates of elements of R = {1, 2, 3, 5}
range (R) = set of second coordinates of elements of R = {4, 6, 9}

(3) Arrow diagram: Let R be a relation from A to B. First we draw two bounded figures to represent A and B respectively. We mark the elements of A and B in these figures. For each (a, b) ∈ R, we draw an arrow from a to b.

Example 7. Let A = {1, 2, 3, 4, 5}. Define a relation R from A to A by R = {(x, y) : y = 2x − 3}.
(i) Depict R using an arrow diagram.
(ii) Find dom (R) and range (R).
Answer:
x = 1 ⇒ y = (2 − 3) = −1 ∉ A
x = 2 ⇒ y = (4 − 3) = 1
x = 3 ⇒ y = (6 − 3) = 3
x = 4 ⇒ y = (8 − 3) = 5
x = 5 ⇒ y = (10 − 3) = 7 ∉ A
∴ R = {(2, 1), (3, 3), (4, 5)}
(i) We may depict it by an arrow diagram as shown:

(ii) We have dom (R) = {2, 3, 4} and range (R) = {1, 3, 5}.

Functions

Function

We can visualise a function as a rule which produces new elements out of some given elements. There are many terms such as 'map' or 'mapping' used to denote a function.
Definition: A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.
In other words, a function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element.
If f is a function from A to B and (a, b) ∈ f, then f(a) = b, where b is called the image of a under f and a is called the pre-image of b under f. The function f from A to B is denoted by f : A → B.
dom (f) = A and range (f) ⊆ B. Also, B is called the co-domain of f.

Example 8. Let X = {1, 2, 3, 4}, Y = {1, 4, 9, 16, 25} and f = {(x, y) : x ∈ X, y ∈ Y and y = x²}.
(i) Show that f is a function from X to Y. Find its domain and range.
(ii) Draw a pictorial representation of the above function.
(iii) If A = {2, 3, 4}, find f(A).
Answer:
(i) We have f = {(x, y) : x ∈ X, y ∈ Y and y = x²}
Giving different values to x from the set X and getting the corresponding values of y = x², we get f = {(1, 1), (2, 4), (3, 9), (4, 16)}
Clearly, every element in X has a unique image in Y. Hence f is a function from X to Y.
dom (f) = {1, 2, 3, 4} = X; range (f) = {1, 4, 9, 16} ⊂ Y
Clearly, 25 ∈ Y does not have its pre-image in X.
(ii) A pictorial representation of the above mapping f is given by

(iii) Now, let A = {2, 3, 4}. Then f(2) = 2² = 4, f(3) = 3² = 9 and f(4) = 4² = 16
∴ f(A) = {f(x) : x ∈ A} = {4, 9, 16}

Example 9. Examine each of the following relations given below and state in each case, giving reasons, whether it is a function or not.
(i) R = {(2, 1), (3, 1), (4, 2)}
(ii) R = {(2, 2), (2, 4), (3, 3), (4, 4)}
(iii) R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}
Answer:
(i) Since 2, 3, 4 are the elements of the domain of R having their unique images, this relation R is a function.
(ii) Since the same first element 2 corresponds to two different images 2 and 4, this relation is not a function.
(iii) Since every element has one and only one image, this relation is a function.

Real Valued Function and Real Function

A function which has either R or one of its subsets as its range is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.

Some Functions and Their Graphs

Identity Function

Let R be the set of real numbers. Define the real valued function f : R → R by y = f(x) = x for each x ∈ R. Such a function is called the identity function. Here the domain and range of f are R. The graph is a straight line. It passes through the origin.

Constant Function

Define the function f : R → R by y = f(x) = c, x ∈ R, where c is a constant. Here the domain of f is R and its range is {c}. The graph is a line parallel to the x-axis. For example, the graph of f(x) = 3 is shown below.

Polynomial Function

A function f : R → R is said to be a polynomial function if for each x in R, y = f(x) = a₀ + a₁x + a₂x² + … + a_nxⁿ, where n is a non-negative integer and a₀, a₁, a₂, …, a_n ∈ R.

Rational Function

Rational functions are functions of the type \( \frac{f(x)}{g(x)} \), where f(x) and g(x) are polynomial functions of x defined in a domain where g(x) ≠ 0.

Modulus Function

The function f : R → R defined by f(x) = |x| for each x ∈ R is called the modulus function. For each non-negative value of x, f(x) is equal to x. But for negative values of x, the value of f(x) is the negative of the value of x, i.e.
\( f(x) = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases} \)
The graph of the modulus function is given by:

Signum Function

The function f : R → R defined by
\( f(x) = \begin{cases} 1, & \text{if } x > 0 \\ 0, & \text{if } x = 0 \\ -1, & \text{if } x < 0 \end{cases} \)
is called the signum function. The domain of the signum function is R and the range is the set {−1, 0, 1}. The graph of the signum function is given by:

Greatest Integer Function

The function f : R → R defined by f(x) = [x], x ∈ R, assumes the value of the greatest integer less than or equal to x. Such a function is called the greatest integer function.
From the definition of [x], we can see that
[x] = −1 for −1 ≤ x < 0
[x] = 0 for 0 ≤ x < 1
[x] = 1 for 1 ≤ x < 2
[x] = 2 for 2 ≤ x < 3 and so on.
The graph of the function is given by: