Quick Revision Notes

Class 11 Mathematics

Chapter: Relations and Functions

1. Cartesian Product of Sets

• Definition: Given two non-empty sets $A$ and $B$, the Cartesian product $A \times B$ is the set of all ordered pairs of elements from $A$ and $B$.
  • A x B = {(a, b) : a ∈ A, b ∈ B}
• Key Properties:
  • If $n(A) = p$ and $n(B) = q$, then $n(A \times B) = pq$.
  • $A \times B = \phi$ if either $A$ or $B$ is a null set.
  • $A \times B \neq B \times A$ (unless $A = B$ or $A = \phi$ or $B = \phi$).
  • $A \times (B \cup C) = (A \times B) \cup (A \times C)$
  • $A \times (B \cap C) = (A \times B) \cap (A \times C)$

2. Relations (संबंध)

• Definition: A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of the Cartesian product $A \times B$.
  • R ⊆ A x B
• Domain (प्रांत): The set of all first elements of the ordered pairs in a relation $R$.
  • Domain(R) = {a ∈ A : (a, b) ∈ R for some b ∈ B}
• Range (सarिसar / परिसर): The set of all second elements of the ordered pairs in a relation $R$.
  • Range(R) = {b ∈ B : (a, b) ∈ R for some a ∈ A}
• Codomain (सह-प्रांत): The entire set $B$ is called the codomain of relation $R$. (Note: $\text{Range} \subseteq \text{Codomain}$).
• Total Number of Relations: If $n(A) = p$ and $n(B) = q$, the total number of relations from $A$ to $B$ is $2^{pq}$.

3. Functions (फलन)

• 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 $B$.
  • Notation: $f: A \to B$
  • For every $a \in A$, there exists a unique $b \in B$ such that $(a, b) \in f$, written as $f(a) = b$.
• Important Terms:
  • Domain: Set $A$
  • Codomain: Set $B$
  • Range: {f(a) : a ∈ A} ⊆ B

4. Real Valued Functions and Their Graphs

• Real Function: A function which has either $\mathbb{R}$ or one of its subsets as its range. If its domain is also $\mathbb{R}$ or a subset of $\mathbb{R}$, it is called a Real Valued Function.

Important Standard Functions:

1. Identity Function: $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x$ for all $x \in \mathbb{R}$.
   • $\text{Domain} = \mathbb{R}$, $\text{Range} = \mathbb{R}$
2. Constant Function: $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = c$, where $c$ is a constant.
   • $\text{Domain} = \mathbb{R}$, $\text{Range} = {c}$
3. Polynomial Function: $f(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$, where $n$ is a non-negative integer and $a_i \in \mathbb{R}$.
4. Rational Functions: 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 $g(x) \neq 0$.
5. Modulus Function (मापॉक फलन): $f: \mathbb{R} \to \mathbb{R}$ defined by
   • f(x) = |x| = { x, if x ≥ 0 ; -x, if x < 0 }
   • $\text{Domain} = \mathbb{R}$, $\text{Range} = [0, \infty)$
6. Signum Function (सिग्नम फलन): $f: \mathbb{R} \to \mathbb{R}$ defined by
   • f(x) = { 1 if x > 0 ; 0 if x = 0 ; -1 if x < 0 }
   • $\text{Domain} = \mathbb{R}$, $\text{Range} = {-1, 0, 1}$
7. Greatest Integer Function (महत्तम पूर्णांक फलन): $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = [x]$, where $[x]$ is the greatest integer less than or equal to $x$.
   • $\text{Domain} = \mathbb{R}$, $\text{Range} = \mathbb{Z}$ (Integers)

5. Algebra of Real Functions

For any two real functions $f: X \to \mathbb{R}$ and $g: X \to \mathbb{R}$:

1. Addition: $(f + g)(x) = f(x) + g(x)$
2. Subtraction: $(f - g)(x) = f(x) - g(x)$
3. Scalar Multiplication: $(cf)(x) = c \cdot f(x)$ (where $c$ is a real number)
4. Multiplication (Pointwise): $(f \cdot g)(x) = f(x) \cdot g(x)$
5. Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (provided $g(x) \neq 0$)