Reflexive Relation

Duration: 7 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:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video lecture provides a detailed mathematical exploration of relations, specifically focusing on the properties of reflexive relations. It begins by establishing the foundational counting principle: for a set A with n elements, the Cartesian product AxA has n^2 elements, leading to a total of 2^(n*n) possible relations. The instructor then defines a reflexive relation, requiring that every element x in A must be related to itself, denoted as (x, x) being in R. Through a series of examples and a structured table of questions, the lecture demonstrates how to identify reflexive relations on a set like {1, 2, 3}. The session concludes by deriving the formula for the total number of reflexive relations, which is 2^(n^2 - n), and discussing key properties such as the behavior of unions, intersections, and inverses of reflexive relations.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor begins by reviewing the total number of relations possible on a set. He explains that if a set A has n elements, the Cartesian product AxA will contain n^2 elements or pairs. Based on this, he states that the total number of possible relations is 2 raised to the power of n times n. He writes "A x A" and "n^2" on the board to illustrate the set of pairs. He briefly writes "A x B" but crosses it out to emphasize that the discussion is about relations on a single set A. This sets the stage for understanding the vast number of potential relations before narrowing down to specific types.

  2. 2:00 – 5:00 02:00-05:00

    The focus shifts to the definition of a "Reflexive relation". The instructor writes that a relation R on a set A is reflexive if for all x in A, the pair (x, x) is in R. To visualize this, he uses a 3x3 table for the set A={1, 2, 3}, highlighting the diagonal cells (1,1), (2,2), and (3,3) as the required elements. A table of questions appears, asking students to identify which of the listed relations are reflexive or irreflexive. The instructor marks the first row (AxA) as reflexive with a checkmark and the second row (empty set) as not reflexive with an X. He continues to evaluate the third relation {(1,1), (2,2), (3,3)} as reflexive.

  3. 5:00 – 6:59 05:00-06:59

    The instructor delves into the properties and counting of reflexive relations. He identifies the smallest reflexive relation as the diagonal set Delta A, containing only {(1,1), (2,2), (3,3)}, and the largest as the full Cartesian product AxA. He derives the formula for the total number of reflexive relations as 2^(n^2 - n), explaining that n elements are fixed on the diagonal, leaving n^2 - n elements to be chosen freely. He addresses true/false questions, confirming that the union and intersection of reflexive relations are reflexive, as are supersets and inverses. A binary table for A={a,b} is shown to visually demonstrate the 16 possible relations and highlight the 4 that are reflexive.

This lecture systematically builds the concept of reflexive relations from basic counting principles to specific properties. It starts by establishing the total space of relations (2^(n^2)) and then narrows the focus to the subset of reflexive relations. The instructor uses visual aids like tables and diagonal markings to make the abstract definition concrete. The derivation of the formula 2^(n^2 - n) is a key takeaway, showing how the fixed diagonal elements reduce the degrees of freedom. Finally, the discussion on properties like union, intersection, and inverses provides a complete algebraic understanding of how reflexive relations behave, preparing students for more complex relation problems.

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