11.5 Practice Question

Duration: 4 min

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Inside: a video lesson and guided study material.

Module outline

  1. Discrete Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic
  2. DataBase Management System/DBMS: 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
  3. Digital Electronics: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  4. Computer Architecture: Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design
  5. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  6. C Language: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  7. Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. 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
  10. Theory Of Computation/Automata Theory: 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
  11. 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
  12. Engineering Mathematics: Permutation and Combination, Linear Algebra, Calculus, Probability, Statistics
  13. General Aptitude: Ratio and Proportion (Ratios), Divisibility Rules, Data Interpretation, Logarithm, Number System, HCF LCM, Sequence and Series (Series), Speed Time and Distance, Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Data Sufficiency, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Percentage, Mensuration and Geometry, Mental Ability, Arithmetic, Profit and Loss, Powers and Exponents (Surds and Indices), Average, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Syllogisms, Venn Diagram, Seating Arrangements, Blood Relations, Directions (Direction Test), Analogy, Algebra, Time and Work, Analytical Reasoning (Counting Figures Reasoning), Puzzle Solving (Puzzles), Cubes & Dices, Ranking, Order and Sequence, Mixture and Alligation, Age Problems, Clock, Selection Decision Table (Decision Making), Data Arrangement
  14. English (Verbal Aptitude): Vocabulary, Noun, Subject Verb Agreement (Verb Noun Agreement), Adjectives, Tenses, Pronoun, Preposition, Direct and Indirect Speech, Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Completion (Fill in the blanks), Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Correction (Error Correction), Verbal Analogy (Word Based Analogy), Conjunction, Interjection, Verb, Articles, Adverb, Modals, Sentence Construction
  15. Live Classes Recordings(Earlier Batch): GATE 2026 Live Class
  16. Full Mock Test:
  17. Previous Year Papers:
  18. GATE 2026 Counselling: Counselling and Guidance Sessions
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video segment presents a practice problem on abstract algebra, specifically focusing on identifying which mathematical structures fail to satisfy the group axioms under a given operation. The instructor guides students through analyzing four multiple-choice options involving sets of integers, multiples, powers of 2, and complex numbers. The core pedagogical objective is to test the understanding of closure, associativity, identity, and inverse properties. The video demonstrates a methodical approach to verifying group structures by checking specific axioms, particularly the existence of an inverse element for every member in the set. The instructor uses visual annotations on screen to highlight critical failures, such as the lack of a multiplicative inverse for zero in the set of complex numbers.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a multiple-choice question displayed on screen asking to identify which set and operation do not form a group. The options listed are: (a) even integers under addition, (b) multiples of k under addition, (c) powers of 2 under multiplication, and (d) the set of complex numbers under multiplication. The instructor begins by analyzing option (d), writing '0 x [] = 1' on the screen to demonstrate that zero has no multiplicative inverse. This visual proof highlights a failure of the inverse axiom, leading the instructor to mark option (d) with a cross as the correct answer. The annotation '0' is written below the complex number set to emphasize the problematic element.

  2. 2:00 – 4:17 02:00-04:17

    The instructor shifts focus to option (c), the set of powers of 2 under multiplication, denoted as {2^n, n ∈ Z}. He writes out the elements explicitly as ..., 1/4, 1/2, 1, 2, 4, ... to illustrate the set's composition. To verify closure, he writes the equation '2^a x 2^b = 2^(a+b)', demonstrating that multiplying two powers of 2 results in another power of 2, satisfying the closure property. He circles specific elements like '2^2' and writes '2^3' to show the multiplication process. The instructor also revisits option (a) and (b), crossing them out to indicate they are valid groups, while continuing to analyze the inverse property for option (c) by checking if every element has a reciprocal within the set.

The lecture effectively uses a counterexample approach to teach group theory concepts. By systematically eliminating valid groups (options a and b) and identifying the specific axiom failure in option d, the instructor clarifies that the set of complex numbers under multiplication is not a group because it includes zero, which lacks an inverse. The analysis of option c further reinforces the importance of checking all axioms, as the set {2^n} actually forms a group under multiplication due to closure and inverse properties (since 1/2^a is also in the set). The visual evidence of writing equations and crossing out options provides a clear, step-by-step method for students to follow when solving similar problems. The key takeaway is that the presence of an element without an inverse, such as zero in multiplication, immediately disqualifies a set from being a group.

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