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.
[University] Introduction to Abstract Algebra
[University] Introduction to Abstract Algebra
Master the foundational structures of modern mathematics. This learning track provides a direct path through abstract algebra, from basic sets to groups, rings, and fields. It delivers the analytical framework essential for advanced theoretical work. This programme is for undergraduate students in mathematics, computer science, or theoretical physics. It is also essential for professionals requiring a rigorous grasp of algebraic structures for work in cryptography, algorithm design, or quantum computing. Construct rigorous proofs and analyse the properties of groups, rings, and fields. This programme directly prepares you for postgraduate studies in pure mathematics and for advanced technical roles in cryptography and algorithm theory.

Master the foundational structures of modern mathematics. This learning track provides a direct path through abstract algebra, from basic sets to groups, rings, and fields. It delivers the analytical framework essential for advanced theoretical work. This programme is for undergraduate students in mathematics, computer science, or theoretical physics. It is also essential for professionals requiring a rigorous grasp of algebraic structures for work in cryptography, algorithm design, or quantum computing. Construct rigorous proofs and analyse the properties of groups, rings, and fields. This programme directly prepares you for postgraduate studies in pure mathematics and for advanced technical roles in cryptography and algorithm theory.

Course Chapters

1. Introduction

Definition of binary operations, signs and symbols, binary operation on a set.

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2. Closure

Definition of closure property, examples of closure in binary operations.

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3. Asssociativity

Definition of associativity, notational conventions, examples of associative operations.

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4. Commutativity

Definition of commutativity, examples of commutative operations.

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5. Distributivity

Definition of distributivity, examples involving one operation distributing over another.

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6. Identity Element

Definition of identity element, examples in various binary operations.

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7. Inverse Element

Definition of inverse element, examples of invertibility under binary operations.

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8. Cayley Tables

Meaning and construction of Cayley tables, Latin square property, related structural concepts.

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