Problems on Closure Property and Algebraic Structure

Duration: 4 min

This video lesson is available to enrolled students.

Return to /learn/COAL-INDIA-MANAGEMENT-TRAINEE/engineering-maths/group-theory/closure-algebraic-structure-associative-semigroup-identity-monoid/asset-problems-on-closure-property-and-algebraic-structure after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. Paper-2 | 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
  3. Paper-2 | Databases / DBMS & SQL: 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
  4. Paper-2 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  5. Paper-2 | Computer Networks & Security: Introduction to CN, Data Communication, 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, Network Access Technologies, Network Security
  6. Paper-2 | Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep
  7. Paper-2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  8. Paper-2 | 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
  9. Paper-2 | Computer Organization & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer
  10. Paper-2 | 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. Paper-1 | General Awareness: Polity, Physics, Biology, Chemistry, Static General Knowledge, GK- Current Affairs, History, Geography, Indian Economy, Environment
  12. Paper-1 | Logical Reasoning: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Data Interpretation, Blood Relations, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Clock, Calendar, Cubes & Dices, Directions (Direction Test), Analogy, Seating Arrangements, Analytical Reasoning (Counting Figures Reasoning)
  13. Paper-1 | Quantitative Aptitude: Ratio and Proportion (Ratios), Mixture and Alligation, Permutation and Combination, Number System, HCF LCM, Speed Time and Distance, Average, Age Problems, Percentage, Profit and Loss, Time and Work, Pipe and Cistern, Simple Interest and Compound Interest, Powers and Exponents (Surds and Indices), Mensuration and Geometry, Probability, Statistics, Arithmetic, Algebra
  14. Paper-1 | English: Vocabulary, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Cloze Test (Cloze Passage)
  15. Coal India Previous Year Paper:
  16. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus
  17. Coal India Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video lecture introduces fundamental concepts of abstract algebra, specifically focusing on closure properties and algebraic structures. The instructor begins by defining the Closure Property: for a non-empty set A and a binary operation *, A is closed if for all a, b in A, the result a*b is also in A. He then defines an Algebraic Structure as a set A that satisfies the closure property with respect to a binary operation *. Throughout the first segment, the instructor works through a comprehensive table on the right side of the screen, evaluating various sets (Natural numbers N, Integers Z, Real numbers R, Even numbers E, Odd numbers O, etc.) against different operations (addition +, subtraction -, multiplication x, division /). He marks red checkmarks for closed sets and red crosses for non-closed sets, providing concrete examples like (N, +) being closed while (N, -) is not. He also writes calculations on the whiteboard, such as 2-3 = -1 and 3/2 = 1.5, to illustrate why certain operations fail closure for specific sets like Natural numbers. He systematically goes down the list, marking (Z, +) and (Z, x) as closed, while (Z, /) is not. He continues with (R, +), (R, -), (R, x) being closed, but (R, /) having issues with zero. He also evaluates (M, +), (M, x), (E, +), (E, x), (O, +), (O, x), (R-0, x), (R-0, /), and (Non-Singular Matrix, x). In the second segment, the lecture transitions to the Associative Property and Semi-Groups. The Associative Property is defined as: for all a, b, c in A, (a*b)*c = a*(b*c). A Semi-Group is then defined as a set that satisfies both the closure property and the associative property. The table on the right is updated with a new column labeled Algebraic Structure (indicating if it's a semi-group). The instructor systematically goes through the list again, marking Y for sets that form a semi-group (like (N, +), (Z, +), (R, +)) and N for those that do not (like (N, -), (Z, -)). He highlights that while addition and multiplication generally form semi-groups on standard number sets, subtraction and division often fail due to lack of associativity or closure. The video concludes with the evaluation of more complex structures like Non-Singular Matrices under multiplication, confirming they form a semi-group. The visual aid of the table serves as a quick reference guide for students to memorize which standard sets and operations form valid algebraic structures. The instructor's methodical approach ensures that students understand the necessary conditions for different algebraic structures by applying them to familiar number systems.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor defines the Closure Property and Algebraic Structure. He uses a table to evaluate sets like (N, +), (N, -), (Z, +), (Z, -) against operations +, -, x, /. He marks checkmarks for closed sets and crosses for non-closed sets. He writes calculations on the board like 2-3 = -1 and 3/2 = 1.5 to demonstrate closure failures.

  2. 2:00 – 3:39 02:00-03:39

    The lecture transitions to the Associative Property and Semi-Group definitions. The table is updated with a new column for Algebraic Structure (Y/N). The instructor marks Y for semi-groups like (N, +), (Z, +), (R, +) and N for non-semi-groups like (N, -), (Z, -). He concludes by evaluating (Non-Singular Matrix, x) as a semi-group.

The lesson progresses from basic closure requirements to the more complex definition of a semi-group. By systematically applying definitions to a standard table of number sets and operations, the instructor provides a clear framework for identifying algebraic structures. The visual table acts as a summary tool, reinforcing that addition and multiplication are generally well-behaved (closed and associative) while subtraction and division often fail these properties.

Loading lesson…