Relations and Functions

RELATIONS AND FUNCTIONS - COMPREHENSIVE CHAPTER NOTES

CARTESIAN PRODUCT OF SETS

**Definition**: Given two non-empty sets P and Q, the Cartesian product P × Q is the set of all ordered pairs (p, q) where p ∈ P and q ∈ Q.

**Notation**: P × Q = {(p, q) : p ∈ P, q ∈ Q}

**Key Points**:

If either P or Q is empty, then P × Q = ∅

Order matters: (p, q) ≠ (q, p) unless p = q

Two ordered pairs (a, b) and (c, d) are equal if and only if a = c AND b = d

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

If A and B are non-empty and either is infinite, then A × B is infinite

**Example**: If A = {red, blue} and B = {bag, coat, shirt}, then A × B has 2 × 3 = 6 ordered pairs:

A × B = {(red, bag), (red, coat), (red, shirt), (blue, bag), (blue, coat), (blue, shirt)}

**Important Observation**: P × Q ≠ Q × P in general. The Cartesian product is NOT commutative.

**Practical Applications**:

R × R represents all points in 2D plane: (x, y) where x, y ∈ R

R × R × R represents all points in 3D space: (x, y, z) where x, y, z ∈ R

Vehicle license plates, coordinate systems, coding systems

**Worked Example**:

If (2x + 1, y – 3) = (7, 5), find x and y.

Since ordered pairs are equal:

2x + 1 = 7 ⟹ 2x = 6 ⟹ x = 3

y – 3 = 5 ⟹ y = 8

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 × B.

**Notation**: R ⊆ A × B, written as R: A → B

**Key Terminology**:

**Domain of R**: The set of all first elements of ordered pairs in R. Denoted as Dom(R) ⊆ A

**Range of R**: The set of all second elements of ordered pairs in R. Denoted as Range(R) ⊆ B

**Codomain of R**: The entire set B (not necessarily equal to Range)

**Image**: If (a, b) ∈ R, then b is the image of a

**Preimage**: If (a, b) ∈ R, then a is the preimage of b

**Representations of Relations**:

1. **Roster Form**: List all ordered pairs explicitly

Example: R = {(1,2), (2,3), (3,4)}

2. **Set-Builder Form**: Describe the relationship between elements

Example: R = {(x, y) : y = x + 1, x ∈ A, y ∈ B}

3. **Arrow Diagram**: Visual representation with arrows from domain to range

**Important Fact**: The total number of possible relations from set A to set B = 2^(n(A) × n(B))

If n(A) = p and n(B) = q, total relations = 2^(pq)

**Worked Example**:

Let A = {1, 2, 3, 4, 5, 6}. Define R = {(x, y) : y = x + 1}

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}

**Example with Arrow Diagram**:

If R shows relationship "x is square of y" where P = {4, 9, 25} and Q = {-3, -2, 2, 3, 5}

R = {(4, 2), (4, -2), (9, 3), (9, -3), (25, 5), (25, -5)}

Domain = {4, 9, 25}

Range = {-3, -2, 2, 3, 5}

Codomain = Q = {-3, -2, 2, 3, 5}

FUNCTIONS

**Definition**: A relation f from set A to set B is called a **function** (or mapping) if:

1. Every element of A has an image in B

2. Each element of A has exactly ONE image in B

3. Domain of f = A

**Notation**: f: A → B, or f(a) = b means (a, b) ∈ f

**Key Conditions for Function**:

No element in domain should be unmapped

No element in domain should map to two or more different elements in codomain

If (a, b) ∈ f and (a, c) ∈ f, then b = c

**Distinguishing Function from Relation**: A relation is a function only if it passes the **vertical line test** (graphically) and satisfies the uniqueness condition.

**Worked Example**:

Examine if R = {(1, 2), (2, 3), (3, 4)} is a function:

Each element of domain {1, 2, 3} has exactly one image ✓

This IS a function

Examine if R = {(2, 2), (2, 4), (3, 3)} is a function:

Element 2 maps to both 2 and 4 ✗

This is NOT a function

REAL VALUED AND REAL FUNCTIONS

**Real Valued Function**: A function whose range is either R or a subset of R.

**Real Function**: A function whose domain is either R or a subset of R, AND whose range is either R or a subset of R.

These are critical for calculus and higher mathematics.

SPECIAL FUNCTIONS AND THEIR GRAPHS

Identity Function

**Definition**: f: R → R defined by f(x) = x for all x ∈ R

**Domain**: R

**Range**: R

**Graph**: Straight line through origin at 45° to both axes

**Properties**: f(x) = x is the simplest polynomial function of degree 1

Constant Function

**Definition**: f: R → R defined by f(x) = c for all x ∈ R, where c is a constant

**Domain**: R

**Range**: {c}

**Graph**: Horizontal line parallel to x-axis

**Example**: f(x) = 5 gives a horizontal line at height 5

Polynomial Functions

**Definition**: f: R → R defined by f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ

where n is non-negative integer and a₀, a₁, a₂, ..., aₙ ∈ R

**Degree**: The highest power of x present

**Examples**: f(x) = x² - 3x + 2 (quadratic), f(x) = x³ + 1 (cubic)

**Non-example**: f(x) = x^(1/2) + 2x (not polynomial due to fractional exponent)

Quadratic Function

**Standard form**: f(x) = ax² + bx + c, where a ≠ 0

**Domain**: R

**Range**: Depends on value of a and discriminant

**Graph**: Parabola opening upward (a > 0) or downward (a < 0)

**Vertex**: (-b/2a, f(-b/2a))

**Worked Example**: f(x) = x²

f(-2) = 4, f(-1) = 1, f(0) = 0, f(1) = 1, f(2) = 4

Domain = R, Range = [0, ∞)

Graph: U-shaped parabola with vertex at origin

Cubic Function

**Definition**: f(x) = ax³ + bx² + cx + d, where a ≠ 0

**Domain**: R

**Range**: R (cubic functions always cover all real values)

**Graph**: S-shaped curve passing through origin if f(x) = x³

**Properties**: Odd function (f(-x) = -f(x)) when simplified form is x³

**Worked Example**: f(x) = x³

f(-2) = -8, f(-1) = -1, f(0) = 0, f(1) = 1, f(2) = 8

Domain = R, Range = R

Passes through origin with rotational symmetry

Rational Functions

**Definition**: f(x) = p(x)/q(x) where p(x) and q(x) are polynomials and q(x) ≠ 0

**Domain**: All real numbers except where q(x) = 0

**Example**: f: R - {0} → R defined by f(x) = 1/x

Domain: All real numbers except 0

Range: All real numbers except 0

Graph: Hyperbola with asymptotes at x = 0 and y = 0

Discontinuous at x = 0

Modulus (Absolute Value) Function

**Definition**: f: R → R defined by f(x) = |x|

**Formula**:

f(x) = x when x ≥ 0

f(x) = -x when x < 0

**Domain**: R

**Range**: [0, ∞)

**Graph**: V-shaped curve with vertex at origin

**Properties**: Even function (f(-x) = f(x)), non-differentiable at x = 0

**Worked Example**:

f(-3) = 3, f(-1) = 1, f(0) = 0, f(1) = 1, f(3) = 3

Signum Function

**Definition**: f: R → R defined by

f(x) = 1 if x > 0

f(x) = 0 if x = 0

f(x) = -1 if x < 0

**Domain**: R

**Range**: {-1, 0, 1}

**Graph**: Three horizontal rays (discontinuous at x = 0)

**Applications**: Used to determine sign of expressions

Greatest Integer Function (Floor Function)

**Definition**: f: R → R defined by f(x) = [x], the greatest integer less than or equal to x

**Domain**: R

**Range**: Z (set of integers)

**Properties**:

[x] = n if n ≤ x < n+1 for integer n

[x] = -1 for -1 ≤ x < 0

[x] = 0 for 0 ≤ x < 1

[x] = 1 for 1 ≤ x < 2

**Graph**: Step function with jumps at integer values

**Worked Examples**:

[2.3] = 2, [5] = 5, [-2.7] = -3, [0.9] = 0

ALGEBRA OF REAL FUNCTIONS

**Prerequisites**: Let f: X → R and g: X → R be two real functions, where X ⊆ R

Addition of Functions

**Definition**: (f + g): X → R defined by (f + g)(x) = f(x) + g(x) for all x ∈ X

**Domain**: X (intersection of domains of f and g)

**Example**: If f(x) = x² and g(x) = x, then (f + g)(x) = x² + x

Subtraction of Functions

**Definition**: (f - g): X → R defined by (f - g)(x) = f(x) - g(x) for all x ∈ X

**Domain**: X

**Example**: If f(x) = 2x and g(x) = 1, then (f - g)(x) = 2x - 1

Scalar Multiplication

**Definition**: (αf): X → R defined by (αf)(x) = α·f(x) for all x ∈ X, where α ∈ R is a scalar

**Domain**: X

**Example**: If f(x) = x² and α = 3, then (3f)(x) = 3x²

Multiplication of Functions

**Definition**: (f·g): X → R defined by (f·g)(x) = f(x)·g(x) for all x ∈ X

**Also called**: Pointwise multiplication

**Domain**: X

**Example**: If f(x) = x and g(x) = x + 1, then (f·g)(x) = x(x + 1) = x² + x

Division of Functions

**Definition**: (f/g): X → R defined by (f/g)(x) = f(x)/g(x) for all x ∈ X where g(x) ≠ 0

**Domain**: {x ∈ X : g(x) ≠ 0}

**Restriction**: g(x) must never be zero in the domain

**Example**: If f(x) = 1 and g(x) = x, then (f/g)(x) = 1/x with domain R - {0}

**Important Note**: When combining functions, always ensure that the resulting function is well-defined. The domain of (f/g) must exclude points where g(x) = 0.

KEY EXAM POINTS TO REMEMBER

1. **Ordered Pairs**: Always respect order; (a, b) ≠ (b, a) unless a = b

2. **Function Definition**: EVERY element of domain must have exactly ONE image

3. **Domain vs Codomain**: Domain is what we use; Codomain includes unused elements

4. **Special Functions**: Know graphs and properties of f(x) = x, x², x³, 1/x, |x|, [x], sgn(x)

5. **Number of Relations**: From A to B = 2^(n(A) × n(B))

6. **Composite Properties**: Know intersection and union properties with Cartesian products

7. **Common Mistakes**:

Confusing domain with codomain

Thinking P × Q = Q × P

Forgetting that [x] returns INTEGER, not decimal part

Not checking if function is well-defined at all domain points