Questions
3–5 questions in board exams
Difficulty
Medium
Importance
Core — never skip
Overview
Relations and Functions form the foundational layer of higher mathematics, bridging set theory and calculus. For board exams and competitive entrance tests, mastering the classification of mappings and the invertibility of functions is essential for securing marks in the initial section of the paper.
Types of Relations
A relation R on set A is a subset of A × A. To solve exam questions, you must systematically check for reflexivity, symmetry, and transitivity to identify Equivalence Relations.
- Reflexive: (a, a) belongs to R for all a in A
- Symmetric: (a, b) in R implies (b, a) in R
- Transitive: (a, b) in R and (b, c) in R implies (a, c) in R
- Equivalence Relation: Satisfies all three properties
- Equivalence Class: The set of all elements related to a particular element a
One-One and Onto Functions
Functions are categorized based on their mapping behavior between domain and codomain. Understanding the distinction between injective, surjective, and bijective functions is critical for determining if a function has an inverse.
- One-One (Injective): f(x1) = f(x2) implies x1 = x2
- Onto (Surjective): Range equals Codomain
- Bijective: Both One-One and Onto
- Horizontal Line Test: Use graphically to determine if a function is one-one
- Vertical Line Test: Confirms if a relation is a function
Composition and Inverse
Composition of functions (fog) involves sequential mapping, while finding the inverse requires checking for bijectivity. Exam problems often test the interaction between these operations.
- Composition: (fog)(x) = f(g(x))
- Invertibility: A function is invertible if and only if it is bijective
- Inverse property: fof^-1 = I (identity function)
- (fog)^-1 = g^-1 o f^-1
- Composition is generally not commutative: fog != gof
Formula Sheet
f(x1) = f(x2) => x1 = x2 (One-One)
Range = Codomain (Onto)
(fog)(x) = f(g(x))
(fog)^-1 = g^-1 o f^-1
Exam Tip
Always start proving one-one by assuming f(x1) = f(x2) and aim to arrive at x1 = x2; it is the most robust method for board exam marking schemes.
Common Mistakes
- Failing to check transitivity carefully by assuming that if (a, b) and (b, c) don't exist, the relation isn't transitive (vacuously true)
- Confusing codomain with range when proving onto functions, leading to incomplete proofs
- Assuming every one-one function is necessarily onto without verifying the codomain
More Revision Notes
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