Number of Injective Functions are Possible

Duration: 4 min

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  3. Paper 1 | General Hindi: हिन्दी की रूप रचना, वर्णमाला, समास, बहु-विषयक सामग्री (विभिन्न अवधारणाओं को जोड़ने वाली सामग्री), हिन्दी की ध्वनि और स्वनिम व्यवस्था, प्राचीन और मध्यकालीन भारतीय आर्यभाषाएँ, काव्य के लक्षण, हेतु और प्रयोजन, हिन्दी शब्द रचना, हिन्दी साहित्येतिहास दर्शन, भक्तिकालीन काव्यधाराएँ, भारतीय काव्यशास्त्र की परंपरा, भारतीय नवजागरण और स्वाधीनता आंदोलन, अन्य गद्य विधाएँ एवं प्रवासी साहित्य, निर्धारित उपन्यासों का तुलनात्मक अध्ययन, गोदान, हिन्दी कहानी का विकास, स्वातंत्र्योत्तर और नयी कहानी, हिन्दी निबंध का विकास, शुक्लोत्तर निबंध, द्विवेदी और शुक्ल युग के निबंध, हिन्दी की संवैधानिक स्थिति, आधुनिक भारतीय आर्यभाषाएँ और हिन्दी का उद्भव, हिन्दी भाषा प्रयोग के विविध रूप, रीतिकाल, जीवनी, द्विवेदी युग और छायावाद, रीति, वक्रोक्ति और औचित्य सिद्धान्त, शब्दशक्ति और ध्वनि सिद्धान्त, रस सिद्धान्त और साधारणीकरण, उत्तर आधुनिकतावाद, लोहिया दर्शन, हिन्दी उपन्यास का विकास, हिन्दी नवजागरण, अंबेडकर दर्शन, आदिकाल, प्रयोगवाद और नयी कविता, छायावादी काव्य: प्रसाद और निराला, प्रगतिवाद से समकालीन कविता, निर्गुण भक्ति काव्य, द्विवेदीयुगीन काव्य, रीतिकालीन काव्य, आरंभिक हिन्दी उपन्यास, स्वातंत्र्योत्तर उपन्यास, आत्मकथा, स्वाधीनता-पूर्व प्रेमचंदोत्तर उपन्यास, निर्धारित कहानियों का तुलनात्मक अध्ययन, साठोत्तरी नाटक, प्रसाद और प्रसादोत्तर युग के नाटक, भारतेन्दु युगीन नाटक, स्वातंत्र्योत्तर नाटक, भारतेन्दु युगीन निबंध, संस्मरण और रेखाचित्र, व्यंग्य, एक साहित्यिक की डायरी, संस्कृति के चार अध्याय, सगुण भक्ति काव्य, हिन्दी नाटक, एकांकी और रंगमंच, आधुनिक गद्य और हिन्दी नवजागरण, अलंकार सिद्धान्त, हिन्दी आलोचना का विकास, भक्ति आंदोलन, कल्पना, बिम्ब, प्रतीक, मिथक और फंतासी, नयी कविता का विस्तार और साठोत्तरी कविता, उत्तर-छायावादी और प्रगतिवादी काव्य, काव्यगुण और काव्यदोष, प्लेटो, अरस्तू और पाश्चात्य परंपरा, मार्क्सवाद, स्वच्छन्दतावादी काव्यशास्त्र, आधुनिक पाश्चात्य आलोचना, भाषा और भाषा परिवार, देवनागरी लिपि
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  21. Paper 2 | Soft Computing: Fuzzy Sets, Artificial Neural Networks, Genetic Algorithms
  22. Paper 2 | Distributed System: Distributed Systems
  23. UPPSC Live Class Recordings: UPPSC Polytechnic Lecturer 2025 (CS)
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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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