Relations and Functions

1. Cartesian Product of Sets

Ordered Pair

An ordered pair consists of two objects or elements in a given fixed order.

• Notation: $(a, b)$
• Equality: Two ordered pairs $(a, b)$ and $(c, d)$ are equal if and only if $a = c$ and $b = d$.

Definition of Cartesian Product

Given two non-empty sets $P$ and $Q$. The cartesian product $P \times Q$ is the set of all ordered pairs of elements from $P$ and $Q$. $$P \times Q = \{ (p, q) : p \in P, q \in Q \}$$

Important Note:

• If $n(A) = p$ and $n(B) = q$, then $n(A \times B) = pq$.
• If at least one of $A$ or $B$ is infinite, then $A \times B$ is infinite.
• $A \times A \times A = \{(a, b, c) : a, b, c \in A\}$. This is called an ordered triplet.

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2. Relations

A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of the cartesian product $A \times B$.

• The subset is derived by describing a relationship between the first element ($x$) and the second element ($y$) of the ordered pair $(x, y)$.
• The second element is called the image of the first element.

Domain and Range

1. Domain: The set of all first elements of the ordered pairs in a relation $R$.
2. Range: The set of all second elements of the ordered pairs in a relation $R$.
3. Codomain: The whole set $B$ is called the codomain of the relation $R$.
   • Note: Range $\subseteq$ Codomain.

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3. Types of Relations (Board Exam Priority)

A relation $R$ in a set $A$ is a subset of $A \times A$.

(A) Empty Relation

A relation $R$ in a set $A$ is called empty relation, if no element of $A$ is related to any element of $A$, i.e., $R = \phi \subset A \times A$.

(B) Universal Relation

A relation $R$ in a set $A$ is called universal relation, if each element of $A$ is related to every element of $A$, i.e., $R = A \times A$.

(C) Reflexive Relation

A relation $R$ in a set $A$ is called reflexive if $(a, a) \in R$ for every $a \in A$.

(D) Symmetric Relation

A relation $R$ in a set $A$ is called symmetric if $(a, b) \in R$ implies that $(b, a) \in R$, for all $a, b \in A$.

(E) Transitive Relation

A relation $R$ in a set $A$ is called transitive if $(a, b) \in R$ and $(b, c) \in R$ implies that $(a, c) \in R$, for all $a, b, c \in A$.

(F) Equivalence Relation

A relation $R$ in a set $A$ is said to be an equivalence relation if $R$ is reflexive, symmetric and transitive.

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4. Functions

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 $B$.

Key Conditions:

1. All elements of $A$ must be mapped.
2. Uniqueness: No element in $A$ can have more than one image in $B$.

Notation: If $f$ is a function from $A$ to $B$ and $(x, y) \in f$, then $f(x) = y$.

• $y$ is the image of $x$ under $f$.
• $x$ is the pre-image of $y$.

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5. Some Real Functions and their Graphs

(A) Identity Function

$f: \mathbb{R} \to \mathbb{R}$ defined by $y = f(x) = x$ for each $x \in \mathbb{R}$.

• Graph: A straight line passing through origin with slope 1 ($45^\circ$).
• Domain: $\mathbb{R}$
• Range: $\mathbb{R}$

(B) Constant Function

$f: \mathbb{R} \to \mathbb{R}$ defined by $y = f(x) = c$, where $c$ is a constant.

• Graph: A line parallel to the x-axis.
• Domain: $\mathbb{R}$
• Range: $\{c\}$

(C) Polynomial Function

$f: \mathbb{R} \to \mathbb{R}$ defined by $y = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$.

• Example: Parabola $f(x) = x^2$.
• Domain: $\mathbb{R}$, Range: $[0, \infty)$.

(D) Rational Function

Functions of the type $\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are polynomial functions and $g(x) \neq 0$.

• Example: $f(x) = \frac{1}{x}$.
• Domain: $\mathbb{R} - \{0\}$
• Range: $\mathbb{R} - \{0\}$

(E) Modulus Function (Absolute Value)

$f(x) = |x|$. $$|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$

• Graph: V-shaped starting from origin.
• Domain: $\mathbb{R}$
• Range: $[0, \infty)$ (Non-negative real numbers)

(F) Signum Function

$f(x) = \frac{|x|}{x}$ for $x \neq 0$ and $0$ for $x=0$. $$f(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}$$

• Domain: $\mathbb{R}$
• Range: $\{-1, 0, 1\}$

(G) Greatest Integer Function

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = [x]$, where $[x]$ assumes the value of the greatest integer less than or equal to $x$.

• Example: $[2.3] = 2, [-1.4] = -2$.
• Graph: Step function.
• Domain: $\mathbb{R}$
• Range: $\mathbb{Z}$ (Integers)

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6. Algebra of Real Functions

Let $f: X \to \mathbb{R}$ and $g: X \to \mathbb{R}$ be two real functions.

1. Addition: $(f + g)(x) = f(x) + g(x)$
2. Subtraction: $(f - g)(x) = f(x) - g(x)$
3. Multiplication: $(fg)(x) = f(x)g(x)$
4. Quotient: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, provided $g(x) \neq 0$.

Solved Examples (15+) — Relations and Functions

Example 1

If $(x+1, y-2) = (3, 1)$, find the values of $x$ and $y$.

Solution: Since ordered pairs are equal: $x + 1 = 3 \implies x = 2$. $y - 2 = 1 \implies y = 3$. Ans: $x=2, y=3$.

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Example 2

If $P = \{a, b, c\}$ and $Q = \{r\}$, form the set $P \times Q$ and $Q \times P$.

Solution: $P \times Q = \{(a,r), (b,r), (c,r)\}$. $Q \times P = \{(r,a), (r,b), (r,c)\}$. Note that $P \times Q \neq Q \times P$.

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Example 3

Let $A = \{1, 2, 3\}$. Let $R = \{(1,1), (2,2), (3,3), (1,2), (2,3)\}$. Is $R$ reflexive?

Solution: Yes, because $(1,1), (2,2), (3,3)$ are all present in $R$. Every element is related to itself.

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Example 4

For the above relation $R$, is it symmetric?

Solution: No. We have $(1,2) \in R$, but $(2,1) \notin R$.

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Example 5

Is the relation $R = \{(a,b): a \le b\}$ on $\mathbb{R}$ transitive?

Solution: Let $(a,b) \in R$ and $(b,c) \in R$. This means $a \le b$ and $b \le c$. It implies $a \le c$, so $(a,c) \in R$. Yes, it is transitive.

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Example 6

Let $A = \{1, 2, 3, 4, 5, 6\}$. Define a relation $R$ from $A$ to $A$ by $R = \{(x, y) : y = x + 1\}$. Write down the domain, codomain and range.

Solution: $R = \{(1,2), (2,3), (3,4), (4,5), (5,6)\}$.

• Domain: $\{1, 2, 3, 4, 5\}$.
• Range: $\{2, 3, 4, 5, 6\}$.
• Codomain: $\{1, 2, 3, 4, 5, 6\}$.

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Example 7

Check if $R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}$ on $A=\{1,2,3\}$ is an equivalence relation.

Solution:

1. Reflexive: Yes, $(1,1), (2,2), (3,3)$ present.
2. Symmetric: Yes, $(1,2) \in R \implies (2,1) \in R$.
3. Transitive: $(1,2)$ and $(2,1) \in R \implies (1,1) \in R$. Yes. Since it satisfies all three, it is an equivalence relation.

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Example 8

Find the domain of the function $f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}$.

Solution: Function is defined when denominator $\neq 0$. $x^2 - 8x + 12 = 0 \implies (x-2)(x-6) = 0$. $x = 2, 6$. Domain = $\mathbb{R} - \{2, 6\}$.

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Example 9

Find the domain and range of $f(x) = \sqrt{9 - x^2}$.

Solution: Domain: $9 - x^2 \ge 0 \implies x^2 \le 9 \implies -3 \le x \le 3$. Domain: $[-3, 3]$. Range: Let $y = \sqrt{9-x^2}$. Since square root is non-negative, $y \ge 0$. Also max value is at $x=0$, $y=3$. Range: $[0, 3]$.

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Example 10

Find the domain of $f(x) = \frac{1}{\sqrt{x - 5}}$.

Solution: For square root in denominator: $x - 5 > 0$ (Strictly greater than). $x > 5$. Domain: $(5, \infty)$.

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Example 11

Evaluate $f(x) = x^2 - 3x + 1$ at $x = -1$.

Solution: $f(-1) = (-1)^2 - 3(-1) + 1 = 1 + 3 + 1 = 5$.

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Example 12

Let $f = \{(1,1), (2,3), (0,-1), (-1,-3)\}$ be a function from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $f(x) = ax + b$. Determine $a$ and $b$.

Solution: For $(1,1): 1 = a(1) + b \implies a+b=1$. For $(0,-1): -1 = a(0) + b \implies b=-1$. Substitute $b=-1$ in first eq: $a - 1 = 1 \implies a = 2$. Function is $f(x) = 2x - 1$.

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Example 13

Find the range of $f(x) = 2 - 3\cos x$.

Solution: Range of $\cos x$ is $[-1, 1]$. Max value: $2 - 3(-1) = 5$. Min value: $2 - 3(1) = -1$. Range: $[-1, 5]$.

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Example 14

Find the range of $f(x) = |x - 3|$.

Solution: Modulus function is always non-negative. $|x - 3| \ge 0$. Range: $[0, \infty)$.

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Example 15

Evaluate $[4.7] + [-4.7]$ where [\] is Greatest Integer Function.

Solution: $[4.7] = 4$. $[-4.7] = -5$ (Greatest integer less than -4.7). Sum $= 4 + (-5) = -1$.

Questions and Answers (Board Exam)

Q1. Define an Equivalence Relation.

Answer: A relation $R$ in a set $A$ is said to be an equivalence relation if $R$ is reflexive, symmetric, and transitive.

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Q2. What is the domain of a rational function $f(x) = \frac{p(x)}{q(x)}$?

Answer: The domain is $\mathbb{R} - \{x : q(x) = 0\}$.

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Q3. Draw the graph of the Identity Function.

Answer: It is a straight line passing through the origin $(0,0)$ and making an angle of $45^\circ$ with the positive x-axis ($y=x$).

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Q4. What is the range of the Signum function?

Answer: The range is the set $\{-1, 0, 1\}$.

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Q5. If $n(A) = p$ and $n(B) = q$, how many relations are there from $A$ to $B$?

Answer: Total relations = $2^{pq}$ (Number of subsets of $A \times B$).