Number of Injective Functions are Possible

Duration: 4 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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  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
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  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:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video segment explains the mathematical principles for determining the number of possible functions between two sets, specifically focusing on one-to-one mappings. The instructor begins by presenting the standard permutation formula, $^nP_m = P(n, m)$, as the general method for counting these functions. He highlights a specific scenario where the cardinality of both sets is equal ($|A| = |B| = n$), resulting in $n!$ possible functions. To illustrate the general case, he introduces variables $m$ and $n$ for the sizes of sets A and B respectively. He then constructs a visual example on the whiteboard, defining set A with five distinct elements (1, 2, 3, 4, 5) and set B with six distinct elements (a, b, c, d, e, f). He demonstrates the concept of a function by drawing arrows from elements in set A to elements in set B, showing how each element in the domain must map to exactly one element in the codomain.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor starts by writing the formula for the number of possible functions as $^nP_m = P(n, m)$. He immediately follows this with a special case: if the size of set A equals the size of set B ($|A| = |B| = n$), then the number of functions possible is $n!$. He then begins a practical demonstration by writing $|A| = m$ and $|B| = n$. He draws two vertical ovals representing the sets. Inside the first oval, labeled A, he lists the numbers 1 through 5. Inside the second oval, labeled B, he lists the letters a through f. He draws arrows originating from the elements of set A pointing towards set B to visualize the mapping process. Below the diagram, he writes out the multiplication sequence $6 imes 5 imes 4 imes 3 imes 2$, representing the number of choices available for mapping each element of A to B.

  2. 2:00 – 3:47 02:00-03:47

    Continuing the example, the instructor writes the formal permutation formula $^nP_m = rac{n!}{(n-m)!}$ on the board. He substitutes the specific values from his example, writing $^6P_5 = rac{6!}{(6-5)!}$. He simplifies the denominator to $1!$ and then to $1$, leaving the result as $6!$. He expands $6!$ into its full product form: $6 imes 5 imes 4 imes 3 imes 2 imes 1$. This confirms the manual calculation he performed earlier in the diagram section. He concludes the segment by summarizing the relationship between the set sizes, writing $|A| = m ightarrow |B| = n$ to reinforce the variables used in the permutation formula for one-to-one functions.

The lecture effectively bridges the gap between abstract formulas and concrete counting methods. By starting with the general permutation formula and then applying it to a specific numerical example, the instructor clarifies how to calculate the number of one-to-one functions. The visual aid of drawing sets and arrows helps students understand the underlying logic of the multiplication principle used in the formula. The final expansion of the factorial confirms the consistency between the algebraic formula and the step-by-step counting method.

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