Relations and Functions; Chap 3, Advance Mathematics, Concept Infographics

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NCF 2023 Aligned - Visual Learning Series

Chapter 3: Relations & Functions

Linking variable quantities, mapping inputs to outputs, and mastering the rules of the Cartesian plane.

Core Vocabulary & Definitions

• Ordered Pair (a, b): A pair of elements listed in a rigid sequence where order is significant, such that (a, b) = (c, d) if and only if a = c and b = d.
• Cartesian Product (A × B): The set of all possible ordered pairs (a, b) where the first element belongs to set A and the second belongs to set B.
• Relation (R): Any mathematical rule that pairs elements from set A to set B, formally defined as any subset of the Cartesian product (R is a subset of A × B).
• Function (f): A special relation from set A to set B where every element in domain A is associated with exactly one unique image in codomain B.
• Domain, Range & Codomain: For f: A → B, the Domain is the complete input set A, the Codomain is the target set B, and the Range is the subset of B containing all actual output values.

Essential Formulas & Theorems

Cartesian Cardinality Identity:

n(A × B) = n(A) × n(B)

Example: If set A has 3 elements and set B has 4, then A × B contains 3 × 4 = 12 ordered pairs.

Total Possible Relations:

Total Relations = 2^{n(A) × n(B)}

Example: If n(A) = 3 and n(B) = 2, total possible relations = 2⁶ = 64.

Graph Transformations:

y = f(x) ± c ⇒ Vertical shift by c units

y = f(x + k) ⇒ Horizontal shift by k units

Vertical Line Test:

A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Visual: Cartesian Product (A × B)

Every element of A pairs with every element of B.

Visual: Function vs Non-Function

A function must map every input to exactly ONE output.

Algorithm: Testing if a Relation is a Function

1

Inspect Pre-images

Examine all ordered pairs (x, y) or the arrow diagram. Collect all first components x to analyze domain coverage.

2

Verify Domain

Check if every element in set A is assigned an output. If any element lacks an arrow, it is NOT a function.

3

Test Uniqueness

Ensure no single element in set A maps to two or more distinct values in B. (One-to-Many fails the test).

4

Formulate Domain/Range

If passed, state Domain as full set A and Range as the collection of all unique second components y.

Visual: Vertical Line Test

If a vertical line touches the graph more than once, it is not a function.

Key Properties & Special Functions

• Non-Commutativity of Cartesian Product: In general, A × B is not equal to B × A unless A = B or one set is empty.
• Distributive Operations: A × (B U C) = (A × B) U (A × C).
• Range-Codomain Inclusion: For every valid function f: A → B, the Range is always a subset of the Codomain (Range is a subset of Codomain).
• Modulus Function f(x) = |x|:
  Domain = All Real Numbers (R), Range = [0, infinity).
• Greatest Integer Function f(x) = [x]:
  Domain = All Real Numbers (R), Range = Integers (Z).

Common Pitfalls & Exam Traps

Pitfall 1 (Confusing Many-to-One with One-to-Many): Believing f(x) = x² is not a function because both x = 2 and x = -2 map to y = 4.
Correction: Multiple inputs sharing the same output (Many-to-One) is perfectly valid! A relation fails to be a function only when a single input maps to multiple outputs (One-to-Many).

Pitfall 2 (Ordered Pairs vs Unordered Sets): Assuming (3, 5) = (5, 3) in coordinate plane locations or Cartesian products.
Correction: Ordered pairs strictly enforce position: (a, b) = (c, d) requires both a = c AND b = d. Position matters, unlike sets where {3, 5} = {5, 3}.

Designed by Akash Srivastva | Mathematics Hub - Visual Learning Series

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