Number of Onto (Surjective are Possible)

Duration: 5 min

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Module outline

  1. Discrete Mathematics: Permutation, Combination and Probability / क्रमचय, संचय एवं प्रायिकता, Set Theory, Relations, Functions, Propositional and Predicate Logic, Graph Theory, Group Theory
  2. Digital Logic: Digital Systems & Boolean Basics, Boolean Expression, Boolean Minimization, Logic Gates & Hardware, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep
  3. Computer Organization and Architecture: Basics of COA, Programming the Basic Computer, Cache Memory Organization, Input Output Organisation, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design, Pipelining, Multiprocessors
  4. Programming: Language Design, C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, OOPS with C++
  5. Data Structures: Introduction to DS, Array, Linked List, Stack, Queue, Tree, Graphs, Hashing
  6. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  7. Theory of Computation: Grammar, Regular Language Properties, Moore & Mealy Machines, Turing Machines
  8. 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, Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Pushdown Automata & CFG, Complexity Theory
  9. Operating Systems: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS, Linux OS, Security
  10. Database Management Systems: 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, ORDBMS, NoSQL
  11. Computer Networks: 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 Security
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video lecture focuses on the mathematical concept of counting surjective (onto) functions between two finite sets. The instructor begins by defining the problem of finding the number of onto functions from set A to set B, denoting their cardinalities as $|A| = m$ and $|B| = n$. He introduces the general formula derived from the Principle of Inclusion-Exclusion, which is essential for solving such combinatorial problems. The lecture then transitions to a practical application using a specific question from the GATE 2015 exam, demonstrating how to apply the formula to find the number of onto functions from set $X = \{1, 2, 3, 4\}$ to set $Y = \{a, b, c\}$. The instructor walks through the substitution of values and the step-by-step arithmetic calculation to arrive at the final answer.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor introduces the topic by writing the question 'No of onto function possible from A to B = ?' on the screen. He defines the cardinality of the domain set A as $m$ and the codomain set B as $n$. He then displays the general formula for the number of onto functions: $n^m - {}^nC_1(n-1)^m + {}^nC_2(n-2)^m - {}^nC_3(n-3)^m + \dots + (-1)^n {}^nC_{n-1} 1^m$. This formula is presented as the standard method for calculating surjective functions when the size of the domain is greater than or equal to the size of the codomain.

  2. 2:00 – 4:45 02:00-04:45

    The instructor presents a specific problem from GATE 2015 asking for the number of onto functions from set $X = \{1, 2, 3, 4\}$ to set $Y = \{a, b, c\}$. He identifies $m=4$ and $n=3$ and substitutes these into the previously shown formula. The calculation is written out as $3^4 - {}^3C_1(3-1)^4 + {}^3C_2(3-2)^4 - {}^3C_3(3-3)^4$. He simplifies this to $81 - 3(16) + 3(1)$, performing the arithmetic to get $81 - 48 + 3$, which equals 36. He circles the final answer 36 on the screen to conclude the example.

The video provides a clear pedagogical progression from theory to practice. It starts by establishing the theoretical framework for counting onto functions using the Principle of Inclusion-Exclusion, providing a reusable formula for students. It then immediately reinforces this theory by solving a concrete, exam-style problem. By breaking down the substitution of $m=4$ and $n=3$ into the formula and showing the detailed arithmetic, the instructor ensures students understand not just the formula itself, but also the mechanics of applying it correctly to find the number of surjective mappings between sets of different sizes.

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