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Propositions, Conjunction, Disjunction, Negation, XOR | Lecture 20 | Discrete Structures | CSIT, TUIn this video, we beg...
07/08/2023

Propositions, Conjunction, Disjunction, Negation, XOR | Lecture 20 | Discrete Structures | CSIT, TU

In this video, we begin the chapter of Logic and Proof Methods with a formal look into the idea of Propositions. The different ways of combining propositions lead to "compound propositions" and we look into the operations of Conjunction, Disjunction, Negation and Exclusive OR. We support our discussion with an introduction to truth tables and show the relevant truth tables for the mentioned operations.
The next video shall resume the topic of Implications, and discuss the ideas of the inverse, converse and the contrapositive.

https://www.youtube.com/watch?v=hVQc9_vsVqk&list=PLZDSqgEC7MoI2wDYb1vD3NULdfTf-PgsU&index=20

Note: Chapters and Timestamps are in the description. Help us get them to the video timeline by reaching 1000 subscribers!------Welcome to The Digital Nerds,...

Operations on Large Integers, Zero-One Matrices | Lecture 19 | Discrete Structures | CSIT, TUIn this video, we look into...
07/08/2023

Operations on Large Integers, Zero-One Matrices | Lecture 19 | Discrete Structures | CSIT, TU

In this video, we look into the application of the Chinese Remainder Theorem, and it's usage in enabling modulo arithmetic with a choice of good co-prime moduli set, like 99, 98, 97 and 95. The video demonstrates how writing numbers modulo these remainders helps perform arithmetic with large integers using tuples of appropriate sizes.

The rest of the time in the video is spent discussing the ideas of Zero-One matrices and the various matrix operations: Join, Meet and Boolean Product. A couple of numerical problems are solved relating to the zero-one matrices.

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Chinese Remainder Theorem | Lecture 18 | Discrete Structures | CSIT, TUIn this video, we discuss the idea of system of l...
07/08/2023

Chinese Remainder Theorem | Lecture 18 | Discrete Structures | CSIT, TU

In this video, we discuss the idea of system of linear congruences, and present a method to solve them - the chinese remainder theorem. We walk through the necessary criteria for the theorem to apply, and demonstrate side by side analytical and numerical solutions to a typical problem in the related topic.

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Linear Congruences: Application of Modular Arithmetic | Lecture 17 | Discrete Structures | CSIT, TUIn this video, we dis...
07/08/2023

Linear Congruences: Application of Modular Arithmetic | Lecture 17 | Discrete Structures | CSIT, TU

In this video, we discuss the concepts of Linear Congruences, and explore the process of normalizing a linear congruence by multiplying the coefficient of the equation with it's multiplicative inverse under the same modulus. It then outlines the case where we might want to solve multiple congruences simultaneously, hinting towards the Chinese Remainder Theorem. Then it diverts towards the discussion for Cryptology and Caesar Cipher, keeping the Chinese Remainder Theorem for the next video.

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Integer Operations: Division, Modular Exponentiation | Lecture 16 | Discrete Structures | CSIT, TUIn this video, we esta...
07/08/2023

Integer Operations: Division, Modular Exponentiation | Lecture 16 | Discrete Structures | CSIT, TU

In this video, we estabslish the fundamental concepts for division using the approach of repeated addition, and devise an algorithm for Binary Integer Division using the same approach. Next we look into the concepts of modular exponentiation and how the various properties of modulo congruence can be exploited to simplify large exponents and products. Finally, these concepts are used to devise the modular exponentiation algorithm for any number, and a given binary exponent.

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Binary Integer Operations: Addition, Multiplication | Lecture 15 | Discrete Structures | CSIT, TUIn this video, we follo...
07/08/2023

Binary Integer Operations: Addition, Multiplication | Lecture 15 | Discrete Structures | CSIT, TU

In this video, we follow the algorithm for Binary Integer Addition and Multiplication, as discussed in Discrete Mathematics and its Applications, Kenneth H. Rosen. The idea of these algorithms is to provide a systematic process of iterating through the bits of two given binary numbers and generating a sum and carry - in terms of addition; while in terms of multiplication the algorithm employs a sum of partial products method. Both algorithms are analogous to the pen and paper algorithm of solving the problems.

Note: Chapters and Timestamps are in the description. Help us get them to the video timeline by reaching 1000 subscribers!------Welcome to The Digital Nerds,...

Extended Euclidean Algorithm, Bezout's Coefficients | Lecture 14 | Discrete Structures | CSIT, TUIn this video, we condu...
07/08/2023

Extended Euclidean Algorithm, Bezout's Coefficients | Lecture 14 | Discrete Structures | CSIT, TU

In this video, we conduct a detailed exploration of the Euclidean Algorithm and its fundamental role in number theory and encryption. Starting with a systematic breakdown of the algorithm's application for determining the Greatest Common Divisor (GCD), we aim to provide a clear understanding of its utility and function.

The discussion extends to Bezout's Theorem and its associated coefficients, building upon the context provided by the study of the GCD. We then transition to an examination of the Extended Euclidean Algorithm, specifically focusing on the computation of Bezout's Coefficients using a second reverse pass of the steps from the Euclidean Algorithm.

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Primes, GCD, LCM, Prime Factorization | Lecture 13 | Discrete Structures | CSIT, TUThis video journeys into the fundamen...
07/08/2023

Primes, GCD, LCM, Prime Factorization | Lecture 13 | Discrete Structures | CSIT, TU

This video journeys into the fundamental and exciting world of Primes. Beginning with the basics, we explore the core principles of Primes, setting a strong foundation for the topics that follow. From the mysteries of Euler's Prime Generator to the universality of the Fundamental Theorem of Arithmetic, we dissect these prime concepts with a keen analytical approach.

We then discuss the Greatest Common Divisor (GCD) and its intriguing interplay with Relative Primes, further extending the conversation to Pairwise Relative Primes. Shining light on the concept of the Least Common Multiple (LCM), we embark on an in-depth comparison of GCD and LCM using prime factorization.

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Modulo Arithmetic: Hashing Functions, Random Numbers | Lecture 12 | Discrete Structures | CSIT, TUThis video provides a ...
07/08/2023

Modulo Arithmetic: Hashing Functions, Random Numbers | Lecture 12 | Discrete Structures | CSIT, TU

This video provides a comprehensive exploration of Modular Arithmetic. Starting with the basics, we introduce the Congruence Modulo Notation and elaborate on its real-world implications for solving problems in the realm of congruence. As the discussion advances, we shed light on Congruence Classes and a pivotal theorem in Modular Arithmetic that underscores the preservation of congruence under addition and multiplication.

Next, we take a step towards practical applications, discussing Hashing Functions. These are integral to data management, serving as a method to assign unique identifiers to data units, thus streamlining data retrieval processes.

The topic then pivots to Pseudorandom Number Generators, focusing on the Linear Congruential Generator (LCG). Such generators are crucial in various fields, from computer simulations to cryptographic applications.

In the concluding part of the video, we delve into cycles within the LCG and the importance of parameter selection in optimizing LCG performance. The choices regarding these parameters directly influence the generator's efficiency, a crucial factor in successful computing operations.

Note: Chapters and Timestamps are in the description. Help us get them to the video timeline by reaching 1000 subscribers!------Welcome to The Digital Nerds,...

Divisibility, Division Algorithm, Theorems | Lecture 11 | Discrete Structures | CSIT, TUThis video delves into the mathe...
07/08/2023

Divisibility, Division Algorithm, Theorems | Lecture 11 | Discrete Structures | CSIT, TU

This video delves into the mathematical concept of Divisibility. It starts off by explaining the Divides Notation and how it's used in mathematical expressions. Next, it unpacks the Division Algorithm theorem, followed by an in-depth exploration of three critical theorems on Divisibility. The video takes care to elucidate on the concept of Transitivity in the context of these theorems. The final segment of the video focuses on applying these theories to an analytical problem that involves Divisibility, Factors, and the Floor function. The detailed explanation provided helps in better understanding of these foundational concepts.

Note: Chapters and Timestamps are in the description. Help us get them to the video timeline by reaching 1000 subscribers!------Welcome to The Digital Nerds,...

Summations, Sum of AP & GP, Double Summations | Lecture 10 | Discrete Structures | CSIT, TUThis video begins with Summat...
07/08/2023

Summations, Sum of AP & GP, Double Summations | Lecture 10 | Discrete Structures | CSIT, TU

This video begins with Summations, Seires, distinguishes between summation and series, and introduces the Summation Notation.
It then looks at the sum of some common types of series including arithmetic and geometric series. The convergence criteria of a geometric series is laid out with a graphical demonstration, and some other useful series formulae are introduced. Finally the video explores double summations and some numerical problems on double summations.

Note: Chapters and Timestamps are in the description. Help us get them to the video timeline by reaching 1000 subscribers!------Welcome to The Digital Nerds,...

Sequences: Formal Definition, General Terms| Lecture 9 | Discrete Structures | CSIT, TUThis video discusses the basics o...
07/08/2023

Sequences: Formal Definition, General Terms| Lecture 9 | Discrete Structures | CSIT, TU

This video discusses the basics of sequences and introdues the formal definition of sequences as a function from the domain of Natural Numbers. Then it moves on to explaining the ways of describing sequences - as an index function or a recurrence relation. Next, the general term and basic properties of Arithmetic Sequence and Geometric sequences are stated. Finally, the strong law of small numbers is hinted in context to the same small numbers having the jobs of representing multiple sequences - the famous sequence 1,2,4,8,16,31...

Note: Chapters and Timestamps are in the description. Help us get them to the video timeline by reaching 1000 subscribers!------Welcome to The Digital Nerds,...

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