Symmetric Relation

Duration: 9 min

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  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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  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
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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.

The video provides a comprehensive lecture on symmetric relations within the context of discrete mathematics. The instructor begins by formally defining a symmetric relation: a relation R on a set A is symmetric if for every pair (a, b) in R, the pair (b, a) is also in R. He uses a visual 3x3 matrix to demonstrate this concept, showing how elements must be mirrored across the main diagonal. The lecture then transitions to a practical application where the instructor evaluates a series of specific relations on the set A = {1, 2, 3} to determine their symmetry properties. He systematically fills out a table, marking relations as symmetric or not based on the definition. Finally, the instructor derives the formula for the total number of symmetric relations on a set of size n, which is $2^{n(n+1)/2}$, and discusses related properties involving inverses, unions, intersections, and complements.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor introduces the definition of a symmetric relation, stating that if (a, b) is in R, then (b, a) must also be in R. He draws a 3x3 grid representing a set A = {1, 2, 3} to visualize the concept. He explains that for a relation to be symmetric, the matrix representation must be symmetric across the main diagonal. He circles pairs like (1, 2) and (2, 1) to illustrate that if one exists, the other must exist. He also highlights diagonal elements like (1, 1), (2, 2), and (3, 3), noting that they are their own reverses and do not require a pair. This visual aid helps clarify the condition for symmetry. He emphasizes that the condition applies to all elements in the set A. He uses red ink to draw the matrix and circle the elements, making the visual distinction clear for the students.

  2. 2:00 – 5:00 02:00-05:00

    The lecture moves to a problem-solving session using a table with the set A = {1, 2, 3}. The instructor evaluates seven different relations to check for symmetry. He marks the Cartesian product AxA as symmetric because it contains all possible pairs. The empty set is also marked symmetric. He evaluates specific sets like {(1,1), (2,2), (3,3)} as symmetric. Conversely, he marks {(1,2), (2,3), (1,3)} as not symmetric because the reverse pairs are missing. He derives the formula for the total number of symmetric relations, explaining that diagonal elements can be chosen in $2^n$ ways and off-diagonal pairs in $2^{(n^2-n)/2}$ ways, leading to the total $2^{n(n+1)/2}$. He writes this formula on the board and explains the logic behind the exponent. He also discusses the specific case of n=2 to verify the formula.

  3. 5:00 – 8:30 05:00-08:30

    The instructor completes the analysis of the relations in the table, marking the remaining entries. He summarizes key facts: the smallest symmetric relation is the empty set, and the largest is the Cartesian product AxA. He then addresses theoretical properties. He confirms that if a relation R is symmetric, then $R = R^{-1}$. He validates that the union and intersection of two symmetric relations are always symmetric. However, he refutes the idea that supersets or subsets of a symmetric relation are always symmetric, marking it as false. Finally, he confirms that the complement of a symmetric relation is always symmetric, using a 2x2 example to reinforce the counting logic. He writes True and False next to the statements to clarify the properties. He ensures students understand that while operations like union preserve symmetry, inclusion does not.

The video effectively bridges the gap between the abstract definition of symmetric relations and their practical identification and counting. By starting with a clear definition and a visual matrix representation, the instructor establishes a strong conceptual foundation. The progression to a tabular analysis of specific examples allows students to apply the definition directly, distinguishing between symmetric and non-symmetric cases. The derivation of the counting formula $2^{n(n+1)/2}$ is a crucial takeaway, providing a method to calculate the total number of such relations for any set size. The final section on properties, such as the behavior of unions, intersections, and complements, rounds out the topic by connecting symmetric relations to broader set theory concepts. This structured approach ensures students understand not just what a symmetric relation is, but how to identify it, count it, and manipulate it within mathematical proofs. The use of specific examples like A={1,2,3} and A={a,b} makes the abstract concepts concrete and easier to grasp for exam preparation.

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