Modern algebra
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This document covers key concepts in modern algebra including equivalence relations, partial orders, functions, binary operations, permutation groups, cyclic groups, order of elements, cosets, Lagrange's theorem, normal subgroups, and quotient groups. It provides examples and definitions for each concept. Equivalence relations are defined as reflexive, symmetric, and transitive relations. Partial orders must also be antisymmetric. Functions can be one-to-one or onto. Binary operations satisfy closure and are demonstrated with examples. Permutation groups results are listed regarding even/odd permutations and inverses. Cyclic groups and the order of an element are discussed. Cosets and Lagrange's theorem relate the order of a group to the order of subgroups
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Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Modern algebra
- 1. MODERN ALGEBRA III B.Sc Maths 15UMTC51 K.Anitha.M.Sc.,M.Phil., M.Mohana Malar.M.Sc.,M.Phil.,
- 2. Equivalence Relation A relation R defined on a non empty set S is said to be equivalence relation if it is reflexive, symmetric and transitive. Eg: S=Z, a R b means ab is a perfect square. S=R, a R b means a-b is an integer. S=R, a R b means |a|= |b|.
- 3. Partial order A relation R defined on a set S is said to be antisymmetric if a R b and b R a implies a = b. A relation R in S which is reflexive, antisymmetric and transitive is called a partial order on S. A set S with a partial order R defined on it is called a partially ordered or a poset and is denoted by ( S, R).
- 4. Functions Consider the function f: R Rgiven by f(x)= 3 is called a constant function. A function f: A B is one-one function if distinct elements are in A have distinct image in B under f. A function f: A B is onto function if the range of f is equal to B.
- 5. Binary operations Eg: The usual addition,+ is a binary operation on N,Z,Q,R and C. In any non empty set A,* defined by a*b =a is a binary operation. In( N.*) defined by a*b=aab is a binary operations.
- 6. Permutation Groups Results Any permutation can be expressed as a product of disjoint cycle. Any permutations can be expressed as a product of transpositions. The product of two even permutations is an even permutations. The product of two odd permutations is an even permutations. The product of an even permutations and an odd permutations is an odd permutations. The inverse of an even permutations is an even permutations. The inverse of an odd permutations is an odd permutations. The identity permutations e is an even permutations.
- 7. Cyclic Groups and Order of an element.
- 10. Normal Subgroups and Quotient Groups.