Announcements
Students often seek clear and concise resources to effectively grasp the mathematical concepts
covered in their curriculum. To meet this need, we have prepared this handout, which
offers carefully structured lessons and fully solved exercises aligned with the program set by
the Ministry of Higher Education and Scientific Research.
This document is a comprehensive course handout for Algebra 1, designed to align with the
national curriculum. It is structured into five core chapters, beginning with a foundational
review of Logic and Methods of Reasoning. This first chapter establishes the formal rules
of propositional logic, logical connectors, quantifiers, and essential proof techniques like direct
proof, contraposition, contradiction, and mathematical induction.
The second chapter, Sets and Applications (Maps), covers fundamental set theory, including
operations (union, intersection, complement) and their properties. It then defines the
concept of a map, exploring images, pre-images, and classifying maps as injective, surjective,
or bijective.
Chapter 3, Binary Relations, analyzes relations characterized by reflexivity, symmetry, and
transitivity (equivalence relations, which yield partitions and quotients) and those characterized
by reflexivity, antisymmetry, and transitivity (order relations).
The fourth chapter, Algebraic Structures, provides a systematic development of fundamental
algebraic systems. It begins with internal composition laws, their properties (associativity,
commutativity, distributivity), and the study of special elements (identity, invertible, regular elements).
This foundation leads to the formal definition of groups, including subgroups, cosets,
Lagrange’s Theorem, and group homomorphisms. The chapter further extends to rings and
fields, covering subrings, ideals, ring homomorphisms, and the classification of fields including
finite fields Fp.
Chapter 5 introduces the Ring of Polynomials, constructing R[X] over commutative rings
and examining polynomial arithmetic: divisibility, Euclidean division, GCD computation, and
factorization into irreducibles. The chapter concludes with the theory of polynomial roots, including multiplicity and the Fundamental Theorem of Algebra. Each chapter is supplemented
with comprehensive solved exercises that reinforce the theoretical concepts through practical
application and problem-solving techniques.
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