Equivalence relation practice question

Duration: 6 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
  2. General Awareness: Physics, Chemistry, Biology, History, Geography, Polity, Indian Economy, Environment, Static General Knowledge, GK- Current Affairs, Artificial Intelligence, MS Office Applications, Terms & Abbreviations, Keyboard Shortcuts, Evolution of Indian Banking (Banking & Financial Awareness), Emerging Tech Trends
  3. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Probability, Statistics, Signal Space and Optimum Detection, Probability, Statistics
  4. Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Number Systems and Codes, Registers and Counters, Boolean Algebra and Logic Gates, Microprocessors and Microcontrollers, Logic Families and CMOS Logic, Microprocessors, Logic Gates and Boolean Algebra, Minimization of Boolean Functions, Timing, Hazards and Testing, Logic Families, Combinational Logic Design, Arithmetic Circuits, Waveform Generators and Timer Circuits, Semiconductor Memories and Programmable Logic Devices, Combinational Logic Circuits, Latches and Flip-Flops, Finite State Machines, Sequential Circuits and State Machines, Minimization of Boolean Functions, Arithmetic Circuits, Latches and Flip-Flops, Number Systems and Codes, Registers and Counters
  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
  11. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Security, Windows OS, Distributed Systems, Unix OS, Linux OS
  12. Computer Networks: Introduction to CN, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics, Data Communication, Website, Browser & Services, Web Design & Development, Network Security, Cyber Threats and Malware Attacks, Cyber Safety and Security
  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 lecture demonstrates how to verify whether a relation on the integers is an equivalence relation, then applies that idea to a multiple-choice question about ordered pairs. The instructor defines R1 on Z by (a,b) in R1 iff a+b is even, tests concrete pairs such as (-3,-3), (1,4) X, and (13,11) y, and organizes the proof into rows labeled Z, R, S, T. He shows reflexivity using -3 + (-3) = -6 and the pair (-3,-3), symmetry from a+b = b+a, and transitivity by assuming (a,b) in R and deriving the required condition. A second relation R2, defined by a+b odd, is introduced to contrast with R1 and fails the reflexivity test. The lesson ends with a question asking for the largest and smallest number of ordered pairs in equivalence relations on an n-element set S; option (B) n^2 and n is circled as correct.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor introduces the relation R1 : (a, b) iff (a + b) is even over the set of integers. He underlines (a, b), writes Z beneath it to mark the underlying set, and draws a horizontal integer number line labeled -3 through +3 with arrows. He then begins testing the relation by writing -3 + (-3) = -6 and placing the ordered pair (-3,-3) beside it, showing a concrete instance where the sum is even.

  2. 2:00 – 5:00 02:00-05:00

    The board is organized into rows labeled Z, R, S, and T to structure the equivalence-relation proof. The R row contains examples such as (-3,-3) and (1,4) X, while the S row shows if (a,b) in R then (b,a) in R and the equation a+b = b+a. The instructor writes ig (a,b) in R to begin a formal proof step, using specific integer pairs and the commutativity of addition to demonstrate reflexivity, symmetry, and transitivity for R1.

  3. 5:00 – 6:29 05:00-06:29

    The instructor introduces R2 : (a, b) iff (a + b) is odd over the set of integers to contrast with R1, noting that an odd-sum relation fails reflexivity. The lesson then transitions to a multiple-choice question: Let S be a set of n elements; what are the largest and smallest numbers of ordered pairs in equivalence relations on S? Options (A) n and n, (B) n^2 and n, (C) n^2 and 0, and (D) n and 1 appear, and the instructor circles option (B) n^2 and n as the correct answer.

The central idea is that an equivalence relation must satisfy reflexivity, symmetry, and transitivity. For R1 on Z defined by even sums, the instructor verifies each property: reflexivity is shown because a+a = 2a is always even (illustrated by -3 + (-3) = -6), symmetry follows from a+b = b+a, and transitivity is argued by assuming (a,b) in R1 and deriving the needed condition. The contrasting relation R2, based on odd sums, fails reflexivity because a+a is never odd. The final multiple-choice question connects equivalence relations to ordered pairs: the largest equivalence relation on an n-element set contains all n^2 possible ordered pairs, while the smallest is the identity relation with exactly n ordered pairs. Thus option (B) n^2 and n is correct.

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