Equivalence class & partition of sets

Duration: 4 min

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Module outline

  1. General Intelligence & Reasoning: Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Syllogisms, Venn Diagram, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Data Sufficiency, Puzzle Solving (Puzzles), Data Arrangement, Cubes & Dices, Calendar, Clock, Arithmetic, Number System, HCF LCM, Verbal Analogy (Word Based Analogy), Visual Recognition and Figure Completion, Paper Folding and Cutting, Transformation of Shapes, Clocks and Calendars, Direction Sense, Cubes and Dice, Patterns in Two-Dimensional Figures, Seating Arrangements, Analogy and Classification, Logical Deduction and Induction, Alphabet and Word Tests, Mathematical Operations, Series Completion, Simplification and Approximation, Coding and Decoding, Numerical Reasoning, Assembling and Grouping of Shapes, Three-Dimensional Patterns and Visualization, Vocabulary, Number System, Averages, Mixtures and Alligation, Simple and Compound Interest, Percentages, Algebra, Speed, Time and Distance, Seating and Arrangement Puzzles, Traverse Surveying, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Reading Comprehension, Critical Reasoning, Permutations and Combinations, Narrative Sequencing, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Ratio and Proportion, Time and Work, Mensuration, Data Interpretation, Powers, Surds and Logarithms, Sequences and Series, Articles and Determiners, Sentence Completion, Data Sufficiency, Blood Relations, Simple Interest and Compound Interest, Mixture and Alligation, Sequence and Series (Series), Speed Time and Distance, Algebra, Profit and Loss, Powers and Exponents (Surds and Indices), Permutation and Combination
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  5. Computer Organisation & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer, Register Transfer and Microoperations, Semiconductor Memories and Programmable Logic, Processor Organization and Instructions, Memory Hierarchy and Input/Output, Registers and Counters
  6. Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, HTML Basics, OOPS with C++, Java, Object Oriented Technology
  7. Databases: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, Database Security & Authorization
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory
  10. Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
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  13. Full Length Mock Tests:
  14. Sectional Tests:
  15. Paper 2 - Full-length Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This discrete mathematics lecture introduces equivalence classes and set partitions using a worked example. The instructor presents the set A = {1, 2, 3, 4, 5} and an equivalence relation R = {(1,1),(2,2),(3,3),(4,4),(5,5),(1,4),(4,1),(2,5),(5,2)}, asking students to find the partition of A defined by R. The non-reflexive pairs (1,4), (4,1), (2,5), and (5,2) are highlighted to identify which elements are related. The equivalence class of an element x is defined as [x] = {y | y ∈ A and (x, y) ∈ R}. Applying this definition yields [1] = {1, 4}, [2] = {2, 5}, [3] = {3}, [4] = {1, 4}, and [5] = {2, 5}. The instructor notes that [x] can equal [y] even when x ≠ y, so the partition consists of the distinct classes {1, 4}, {2, 5}, and {3}. A partition is then defined as a subdivision of a set into non-empty, non-overlapping subsets whose union equals the original set and whose intersection is empty.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The problem is displayed on a dark board: “Consider A = {1, 2, 3, 4, 5} an equivalence relation R on A” with R = {(1,1),(2,2),(3,3),(4,4),(5,5),(1,4),(4,1),(2,5),(5,2)} and the instruction “find the partition of a set A, defined by R.” The instructor points to the ordered pairs (1,4), (4,1), (2,5), and (5,2), which are underlined in green as a teaching cue to identify related elements. A vertical list of prompts “[1] =”, “[2] =”, “[3] =”, “[4] =”, and “[5] =” is shown, and the instructor begins writing solutions such as [1] = {1, 4} and [2] = {2, 5}.

  2. 2:00 – 4:05 02:00-04:05

    The completed board work shows [1] = {1, 4}, [2] = {2, 5}, [3] = {3}, [4] = {1, 4}, and [5] = {2, 5}. A definition slide states “Equivalence Class: - of an element is denoted by [x]” and gives “[x] = {y | y ∈ A and (x, y) ∈ R} for all x ∈ A,” with the note “We can have [x] = [y], even if x != y.” The instructor then presents “Partitions of a Set,” displaying the conditions A1 U A2 U ... U An = A and A1 ∩ A2 ∩ ... ∩ An = Φ, explaining that a partition is a subdivision into non-empty, non-overlapping subsets. The distinct equivalence classes {1, 4}, {2, 5}, and {3} are summarized as the partition of A.

The lesson progresses from a concrete problem to general definitions. First, the equivalence relation R on A = {1, 2, 3, 4, 5} is analyzed by identifying the non-reflexive pairs that connect elements: (1,4) and (4,1) link 1 and 4, while (2,5) and (5,2) link 2 and 5. Using the definition [x] = {y | y ∈ A and (x, y) ∈ R}, each equivalence class is computed. The key insight is that equivalent elements share the same class, so [1] = [4] and [2] = [5], reducing the five classes to three distinct subsets. The partition is then formally defined by two conditions: the union of all parts equals A, and their intersection is empty (Φ). This connects equivalence relations to partitions: every equivalence relation induces a unique partition of the set into its distinct equivalence classes.

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