Precalculus 1: Basic notions

• How to solve problems in chosen Precalculus topics (illustrated with 236 solved problems) and why these methods work.
• Repetition of chosen aspects of high school mathematics (basic notions as numbers, functions, sets, equations, and inequalities).
• The basic notions will be defined in a more abstract way, and you will get an insight into their place in Calculus and some other branches of university maths.
• By the end of this course you will be able to understand all the elements in the epsilon-delta definition of limit, both symbols and the content.
• Arithmetic with basic rules (associativity and commutativity of addition and multiplication, distributivity) illustrated for positive integers.
• Basic terminology for equations and inequalities, with examples of linear equations and inequalities, with and without absolute value.
• Basic concepts related to functions (domain, range, graph, surjection, injection, bijection, inverse, composition) and how to work with them.
• Functions: monotone (increasing, decreasing), bounded, even, odd; extremum: maximum, minimum, both local and global.
• Geometrical operations on graphs of functions: translations, reflections, shrinking, dilating. Piecewise functions.
• Absolute value and its role in computing distances, with geometrical illustrations and functional approach.
• Equations of straight lines in the plane: slope and intercept; first glimpse into derivative as the slope of the tangent line, and link to monotonicity.
• Introduction to sequences and series; short notation for sum (Sigma) and product (Pi); arithmetic, geometric, and harmonic progressions.
• Logic: symbols and rules (tautologies) used for creating mathematical statements, definitions, and proofs.
• Set theory: union, intersection, complement, set difference; some important rules and notation.
• Relations: RST (equivalence) relations, order relations, functions as relations.
• Necessary and sufficient conditions: definition and examples.
• Various types of proofs: direct, indirect, by contradiction, induction proof, with some examples.
• Building blocks of mathematical theories: axioms, definitions, theorems, propositions, lemmas, corollaries, etc.
• Decimal expansion of rational and irrational numbers. Density of Q and R\Q in R.
• Cardinality of sets: finite sets, N, Z, Q, R; countable and uncountable sets.
• You will also get a solid background for any future studies in Real Analysis and other courses in university mathematics.

Who this course is for:

• Students who plan to study Calculus, Real Analysis, Discrete Mathematics or Abstract Algebra
• High school students curious about university mathematics; the course is intended for purchase by adults for these students
• Everybody who wants to brush up their high school maths and gain a deeper understanding of the subject
• College and university students studying advanced courses, who want to understand all the details they might have missed in their earlier education

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