📈 RELATIONS AND FUNCTIONS
Complete Mathematics Notes with Examples & Formulas
🧮 CARTESIAN PRODUCT
Definition
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 \( Q \):
\[P \times Q = \{ (p, q) : p \in P, q \in Q \}\]
If either \( P \) or \( Q \) is the null set, then \( P \times Q = \emptyset \).
📊 Example
If \( A = \{ a_1, a_2 \} \) and \( B = \{ b_1, b_2, b_3, b_4 \} \):
\( (a_2, b_1), (a_2, b_2), (a_2, b_3), (a_2, b_4) \} \)
⚖️ Equality Rule
Two ordered pairs are equal if and only if:
First elements are equal AND
Second elements are equal
📏 Cardinality
If \( m(A) = p \) and \( n(B) = q \), then:
\( n(A \times B) = pq \)
∞ Infinite Sets
If \( A \) or \( B \) is infinite, then \( A \times B \) is also infinite.
🔢 Ordered Triplet
\( A \times A = \{ (a, b, c) : a, b, c \in A \} \)
where \( (a, b, c) \) is called an ordered triplet.
🔗 RELATIONS
Definition
A Relation \( R \) from a non-empty set \( A \) to a non-empty set \( B \) is a subset of the cartesian product \( A \times B \).
The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in \( A \times B \).
The second element is called the image of the first element.
DOMAIN
The set of all first elements of the ordered pairs in relation \( R \).
RANGE
The set of all second elements in relation \( R \).
CODOMAIN
The whole set \( B \) is called the codomain of relation \( R \).
📝 Important Notes
• Relation on A: A Relation \( R \) from \( A \) to \( B \) is also stated as a relation on \( A \).
• Total Relations: If \( m(A) = p \) and \( n(B) = q \), then:
Total relations = \( 2^{pq} \)
📐 FUNCTIONS
Definition
A function \( f \) from set \( A \) to set \( B \) is said to be a function if every element of set \( A \) has one and only one image in set \( B \).
Notation
The function \( f \) from \( A \) to \( B \) is denoted by:
\( f: A \to B \)
Image
If \( (a, b) \in f \), then \( b \) is called the image of \( a \) under \( f \).
Preimage
If \( (a, b) \in f \), then \( a \) is called the preimage of \( b \) under \( f \).
📈 Real Valued Function
A function which has either \( \mathbb{R} \) or one of its subsets as its range is called a real valued function.
🌟 TYPES OF FUNCTIONS
Identity Function
Definition: \( f: \mathbb{R} \to \mathbb{R} \) by \( f(x) = x \) for each \( x \in \mathbb{R} \)
\( f(x) = x \)
Constant Function
Definition: \( f: \mathbb{R} \to \mathbb{R} \) by \( f(x) = c \), where \( c \) is constant
\( f(x) = c \)
Domain: \( \mathbb{R} \), Range: \( \{c\} \)
Polynomial Function
Definition: \( f: \mathbb{R} \to \mathbb{R} \) where:
\( f(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n \)
\( n \) is non-negative integer, \( a_i \in \mathbb{R} \)
Rational Function
Definition: Functions of the type \( \frac{f(x)}{g(x)} \), where \( f(x) \) and \( g(x) \) are polynomial functions, \( g(x) \neq 0 \)
\( R(x) = \frac{P(x)}{Q(x)} \)
where \( Q(x) \neq 0 \)
Modulus Function
Definition: \( f: \mathbb{R} \to \mathbb{R} \) by \( f(x) = |x| \)
\( f(x) = |x| \)
Absolute value function
Signum Function
Definition:
\( f(x) = \begin{cases}
1, & x > 0 \\
0, & x = 0 \\
-1, & x < 0
\end{cases} \)
Greatest Integer Function
Definition: \( f: \mathbb{R} \to \mathbb{R} \) by \( f(x) = \lfloor x \rfloor \)
where \( \lfloor x \rfloor \) = greatest integer ≤ \( x \)
\( f(x) = \lfloor x \rfloor \)
Also called floor function
📋 SUMMARY
🧮 Cartesian Product
\( P \times Q = \{(p, q) : p \in P, q \in Q\} \)
Cardinality: \( n(A \times B) = pq \)
🧮 ALGEBRA OF REAL FUNCTIONS
Operations on Functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \)
Function Notation
Let \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) be real-valued functions
The algebra of real functions defines how we can perform arithmetic operations (addition, subtraction, multiplication, division) on functions to create new functions.
Addition of Two Real Functions
Sum of functions \( f \) and \( g \)
Definition
The sum of two real functions \( f \) and \( g \) is defined as:
\[ (f+g)(x) = f(x) + g(x) \]
for all \( x \in \mathbb{R} \)
📝 Example
If \( f(x) = 2x + 3 \) and \( g(x) = x^2 – 1 \), then:
\( (f+g)(x) = (2x + 3) + (x^2 – 1) \) \( = x^2 + 2x + 2 \)
Subtraction of Real Functions
Difference between functions \( f \) and \( g \)
Definition
The difference between two real functions \( f \) and \( g \) is defined as:
\[ (f-g)(x) = f(x) – g(x) \]
for all \( x \in \mathbb{R} \)
📝 Example
If \( f(x) = 3x^2 + 2x \) and \( g(x) = x^2 + 4 \), then:
\( (f-g)(x) = (3x^2 + 2x) – (x^2 + 4) \) \( = 2x^2 + 2x – 4 \)
Multiplication by a Scalar
Scalar multiplication of function \( f \)
Definition
The scalar multiplication of a real function \( f \) by a scalar \( \alpha \) is defined as:
\[ (\alpha f)(x) = \alpha f(x) \]
for all \( x \in \mathbb{R} \)
📝 Example 1
If \( f(x) = 2x + 1 \) and \( \alpha = 3 \), then:
\( (3f)(x) = 3(2x + 1) = 6x + 3 \)
📝 Example 2
If \( f(x) = x^2 \) and \( \alpha = -2 \), then:
\( (-2f)(x) = -2x^2 \)
Multiplication of Two Real Functions
Product of functions \( f \) and \( g \)
Definition
The product of two real functions \( f \) and \( g \) is defined as:
\[ (f \cdot g)(x) = f(x) \cdot g(x) \]
for all \( x \in \mathbb{R} \) (permutation multiplication)
📝 Example
If \( f(x) = x + 2 \) and \( g(x) = x – 3 \), then:
\( (f \cdot g)(x) = (x + 2)(x – 3) \) \( = x^2 – x – 6 \)
Quotient of Two Real Functions
Division of function \( f \) by function \( g \)
Definition
The quotient of two real functions \( f \) and \( g \) is defined as:
\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \]
provided \( g(x) \neq 0 \), \( x \in \mathbb{R} \)
✅ Valid Example
If \( f(x) = x^2 + 1 \) and \( g(x) = x – 1 \), then:
\( \left( \frac{f}{g} \right)(x) = \frac{x^2 + 1}{x – 1} \)
Domain: \( x \neq 1 \)
⚠️ Important Condition
The quotient function is defined only when:
\( g(x) \neq 0 \)
Denominator must not be zero for any \( x \) in domain
📊 SUMMARY OF FUNCTION OPERATIONS
1. Addition
\( (f+g)(x) = f(x) + g(x) \)
Domain: All \( x \in \mathbb{R} \)
2. Subtraction
\( (f-g)(x) = f(x) – g(x) \)
Domain: All \( x \in \mathbb{R} \)
3. Scalar Multiplication
\( (\alpha f)(x) = \alpha f(x) \)
\( \alpha \) is a real number
4. Multiplication
\( (f \cdot g)(x) = f(x) \cdot g(x) \)
Also called pointwise multiplication
5. Quotient
\( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \)
Condition: \( g(x) \neq 0 \)
🌟 KEY PROPERTIES
Commutative Property
\( f + g = g + f \)
\( f \cdot g = g \cdot f \)
Associative Property
\( (f + g) + h = f + (g + h) \)
\( (f \cdot g) \cdot h = f \cdot (g \cdot h) \)
Distributive Property
\( f \cdot (g + h) = f \cdot g + f \cdot h \)
Identity Elements
Zero function: \( f + 0 = f \)
Constant 1: \( f \cdot 1 = f \)
Algebra of Real Functions Complete Notes
Addition • Subtraction • Scalar Multiplication • Multiplication • Quotient
For functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \)