Binary Operations - Introductory Abstract Algebra (Undergraduate Advanced)

This course introduces binary operations???the basic building blocks of algebraic structures. We explore how operations combine elements of a set, and what it means for them to be associative, commutative, or have an identity or inverse. These ideas form the core of how structures like groups and rings are defined. Clear, hands-on, and designed for anyone beginning their journey into abstract algebra.

Enrolment valid for 12 months
This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
MTH 203: Sets, Logic and Algebra I
MTH 203: Sets, Logic and Algebra I
Abstract algebra is the structural foundation of modern advanced mathematics. This track delivers the complete NUC CCMAS MTH 203 curriculum, rigorously transitioning you from computational arithmetic to abstract mathematical reasoning. It provides the necessary prerequisite framework for understanding complex mathematical systems and their applications. This programme is targeted at undergraduate students in mathematics, computer science, and physics requiring a firm grounding in fundamental algebraic structures. It also serves professionals in fields like cryptography or theoretical computer science who need a rigorous theoretical refresher. You will master the precise definitions, properties, and relations of core algebraic structures, including groups, subgroups, rings, and fields. You will gain competence in constructing formal proofs using set theory and logic, and understand how homomorphisms preserve mathematical structure. Completion establishes the critical theoretical base demanded for advanced studies in algebra, coding theory, and algorithm design.

Abstract algebra is the structural foundation of modern advanced mathematics. This track delivers the complete NUC CCMAS MTH 203 curriculum, rigorously transitioning you from computational arithmetic to abstract mathematical reasoning. It provides the necessary prerequisite framework for understanding complex mathematical systems and their applications. This programme is targeted at undergraduate students in mathematics, computer science, and physics requiring a firm grounding in fundamental algebraic structures. It also serves professionals in fields like cryptography or theoretical computer science who need a rigorous theoretical refresher. You will master the precise definitions, properties, and relations of core algebraic structures, including groups, subgroups, rings, and fields. You will gain competence in constructing formal proofs using set theory and logic, and understand how homomorphisms preserve mathematical structure. Completion establishes the critical theoretical base demanded for advanced studies in algebra, coding theory, and algorithm design.

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Course Chapters

1. Introduction
Definition of binary operations, signs and symbols, binary operation on a set.
2. Closure
Definition of closure property, examples of closure in binary operations.
3. Asssociativity
Definition of associativity, notational conventions, examples of associative operations.
4. Commutativity
Definition of commutativity, examples of commutative operations.
5. Distributivity
Definition of distributivity, examples involving one operation distributing over another.
6. Identity Element
Definition of identity element, examples in various binary operations.
7. Inverse Element
Definition of inverse element, examples of invertibility under binary operations.
8. Cayley Tables
Meaning and construction of Cayley tables, Latin square property, related structural concepts.