Least Upper Bound & Greatest Lower Bound

Duration: 7 min

This video lesson is available to enrolled students.

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  3. Paper 1 | General Hindi: भाषा, लिपि, ध्वनि, वर्ण, अक्षर, स्वर, व्यंजन, मात्रा और संयुक्ताक्षर, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, संज्ञा और सर्वनाम, विशेषण और क्रिया, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, वर्णमाला, समास, संधि, Multi-Topic Content (Cross-Concept Material), हिन्दी की ध्वनि और स्वनिम व्यवस्था, प्राचीन और मध्यकालीन भारतीय आर्यभाषाएँ, काव्य के लक्षण, हेतु और प्रयोजन, हिन्दी की रूप रचना, 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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  8. Paper 1 | 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
  9. Paper 1 | 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
  10. Paper 1 | Design and Analysis of Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  11. Paper 1 | Data Structures: Introduction to DS, Array, Linked List, Stack, Queue, Tree, Graphs, Hashing
  12. Paper 1 | Discrete Structures and Theory of Logic: क्रमचय, संचय एवं प्रायिकता / Permutation, Combination and Probability, Set Theory, Relations, Functions, Propositional and Predicate Logic, Graph Theory, Group Theory
  13. Paper 1 | Software Project Management: Software Project Management
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  17. Paper 2 | Image Processing: Digital Image Fundamentals, Image Enhancement, Image Restoration, Color Image Processing, Image Compression
  18. Paper 2 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  19. Paper 2 | Computer Graphics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing
  20. Paper 2 | Web Technologies: Website, Browser & Services, Internet Fundamentals, Web Design & Development, Protocols & Services, Web Publishing & Hosting, Server Side & Security, Emerging Technologies, HTML Basics, JavaScript-Basics, Java
  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)
  24. Previous Year Papers:
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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video is a lecture on Lattice Theory in Discrete Mathematics. The instructor defines Least Upper Bound (LUB/Join/Supremum) and Greatest Lower Bound (GLB/Meet/Infimum). He uses Hasse diagrams to demonstrate how to find these bounds for various subsets of elements within a partially ordered set. He works through multiple examples, calculating LUB and GLB for sets like {g, f, e, c}, {d, f}, {d, e}, {b, c}, {d, c}, and {b, e}. Finally, he introduces the standard notation for these operations: 'V' for Join and '^' for Meet.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor begins the lecture by defining fundamental concepts in lattice theory. The slide displays the title "Least Upper Bound / LUB / Join / Supremum / V" and "Greatest Lower Bound / GLB / Meet / Infimum / ^". He explains that LUB is the "Least value in the upper bound" and GLB is the "Greatest value in the lower bound". He uses a red pen to circle the word "Least" in the title to emphasize the specific nature of the bound being sought. He also underlines the terms "Upper Bound" and "Lower Bound" to distinguish the two main concepts being discussed in this session.

  2. 2:00 – 5:00 02:00-05:00

    The instructor moves to a practical demonstration using a Hasse diagram drawn on a blue background. A table on the right side of the screen lists "Elements" and their corresponding bounds. He analyzes the subset B = {g, f, e, c}, identifying the Upper Bound as {i} and the Least Upper Bound as 'i'. For the Lower Bound, he finds {a, e}, making the Greatest Lower Bound 'e'. He then analyzes B = {d, f}, finding the Upper Bound set {i, g, h, f} and the Least Upper Bound 'f'. For the Lower Bound {a, d}, the Greatest Lower Bound is 'd'. He then switches to a different diagram for B = {d, e}, noting that the Upper Bound is empty (represented by phi) and the Greatest Lower Bound is 'c'. He continues with B = {b, c}, finding the Upper Bound {d, e, f} with LUB 'f' and Lower Bound {a} with GLB 'a'. He also analyzes B = {d, c}, finding LUB 'd' and GLB 'c', and B = {b, e}, finding LUB 'f' and GLB 'a'.

  3. 5:00 – 6:35 05:00-06:35

    In the final segment, the instructor focuses on the algebraic notation for these operations. He writes "a V b" on the white space to represent the Join operation, which corresponds to the Least Upper Bound. He then writes "a ^ b" to represent the Meet operation, corresponding to the Greatest Lower Bound. He emphasizes the symbols V and ^ as the standard mathematical notation used in lattice theory. This notation simplifies the expression of these complex relationships between elements in a partially ordered set, providing a concise way to write LUB(a, b) as a V b and GLB(a, b) as a ^ b. The instructor ensures that students understand the visual representation of these symbols in the context of the diagrams.

The lecture progresses from theoretical definitions to practical application. It starts by defining LUB and GLB with their alternative names (Join/Supremum and Meet/Infimum). The instructor then systematically solves problems using Hasse diagrams, filling out a table to find bounds for different subsets. This practical application solidifies the definitions. Finally, the lesson concludes by formalizing the notation with the symbols V and ^, providing students with the standard algebraic representation for these lattice operations. The video effectively bridges the gap between abstract definitions and concrete calculation methods, ensuring students understand both the conceptual and computational aspects of lattice theory.

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