Precalculus 1: Basic notions

Precalculus 1: Mastering the Foundations for University Mathematics

This comprehensive precalculus course builds a strong mathematical foundation, bridging the gap between high school and university-level studies. You'll develop essential skills in a variety of areas, preparing you for success in calculus and beyond.

Course Overview: Getting Started

• Introduction to the Course: Understand the course structure, learning objectives, and effective study strategies. Learn how to maximize your learning experience.

Essential Mathematical Concepts

• Mathematical Notation: Become fluent in using Greek and Latin letters and a wide range of mathematical symbols crucial for clear and precise communication.
• Numbers and Arithmetic: Explore different types of numbers – natural, integers, rational, irrational, and real – and master fundamental arithmetic operations.
• Absolute Value & Distance: Grasp the concept of absolute value as distance from zero and apply it to measure distances in various contexts, including the Cartesian coordinate system. Learn about reflections and distances in abstract metric spaces.
• Equations & Inequalities: Develop proficiency in solving linear equations and inequalities, including those involving absolute value. Understand solution sets as subsets of R and R².
• Functions: A deep dive into functions! Learn various ways to define functions, analyze their domain and range, identify x and y-intercepts, and explore concepts like surjections, injections, bijections, and inverse functions. Understand increasing/decreasing functions, boundedness, arithmetic operations, compositions, and transformations of graphs, including odd and even functions.
• Logic: Explore the fundamentals of mathematical logic, including symbols, conjunction, disjunction, implication, equivalence, and negation. Learn basic logical rules (tautologies) and proof techniques, including existential and universal quantifiers, and necessary/sufficient conditions.
• Sets: Master set theory basics, including union, intersection, set difference, subsets, and complements. Calculate set cardinality and apply the Inclusion-Exclusion Principle. Explore the connection between set theory and logic.
• Relations: Understand binary relations, with a focus on reflexive, symmetric, and transitive (RST) relations. Learn about equivalence classes and partial order relations.
• Functions as Relations: Explore the definition of a function as a specific type of relation between sets, focusing on domain, codomain, injections, surjections, bijections, and inverse functions.
• Proof Techniques: Learn the meaning and application of axioms, definitions, theorems, lemmas, propositions, and corollaries. Develop skills in various proof methods: direct proof, proof by induction, indirect proof, and proof by contradiction.
• Sequences & Series: Introduce yourself to sequences and series, including arithmetic (AP), geometric (GP), and harmonic (HP) progressions. Learn to use the summation symbols (Sigma and Pi) to represent these concepts.

Important Note: The specific topics covered on your final exam may vary depending on your institution. Always consult with your professor to determine which areas will be assessed.

Resources: A detailed resource file, "001 List_of_all_Videos_and_Problems_Precalculus_1.pdf," provides a comprehensive list of 237 videos and their titles, along with solutions to 236 practice problems. This information is also available within Video 1 ("Introduction to the course").