Questions
4 questions per paper
Difficulty
Medium
Importance
Core — never skip
Overview
Relations and Functions form the foundational layer of higher-order mathematics, mapping how elements in one set relate to another. Understanding these concepts is critical for mastering calculus, as the definition of a function is the gateway to limits, continuity, and differentiability. Aspirants must grasp the Cartesian product and the formal definitions of mapping to excel in these chapters.
Types of Relations
A relation R on a set A is a subset of the Cartesian product A x A. Understanding the properties of reflexivity, symmetry, and transitivity is essential for identifying Equivalence Relations.
- Reflexive: (a, a) belongs to R for all a in A
- Symmetric: If (a, b) in R, then (b, a) in R
- Transitive: If (a, b) in R and (b, c) in R, then (a, c) in R
- Equivalence Relation: Satisfies all three properties
- Identity Relation: R = {(a, a) : a in A}
Domain, Codomain, and Range
These define the operational boundaries of a relation. Domain constitutes all first elements of ordered pairs, while the Range represents the set of all second elements that are actually mapped.
- Domain: Set of all x such that (x, y) belongs to R
- Range: Set of all y such that (x, y) belongs to R
- Codomain: The entire set B in a relation from A to B
- Range is always a subset of the Codomain
- Total number of relations = 2^(n(A) * n(B))
Types of Functions
A function is a special relation where each input has exactly one output. Identifying if a function is one-to-one or onto is the most frequent task in board exams.
- One-to-One (Injective): f(x1) = f(x2) implies x1 = x2
- Onto (Surjective): Range equals Codomain
- Bijective: Both Injective and Surjective
- Many-to-One: More than one x maps to the same y
- Vertical Line Test: Determines if a graph is a function
Formula Sheet
Number of relations = 2^(mn) where m, n are set cardinalities
Condition for injectivity: f(x1) = f(x2) => x1 = x2
Condition for surjectivity: f(x) = y has at least one solution for all y in B
Exam Tip
When proving a function is bijective, always check for both injectivity (algebraically) and surjectivity (by showing range equals codomain) separately to ensure full marks.
Common Mistakes
- Confusing Range with Codomain, failing to solve the inequality for range
- Forgetting to check the Transitive property for all possible triplets (a, b) and (b, c)
- Assuming every relation that passes the vertical line test is also surjective
More Revision Notes
Ready to test yourself?
Play topic-wise Relations and Functions questions in Aspirant Arcade — gamified MCQ practice.
Download Free