Sets, operations with sets, subsets, power set, Cartesian product. Sets of numbers, integers, rational numbers, gentle introduction to real numbers.

A resume of the algebraic properties of rational numbers, ordering in rational numbers. Definition of real numbers through decimal alignment. Absolute value. Intervals. A non-rigorous definition of the separation property or real numbers.

Introduction to functions on arbitrary sets. Image and pre-image. Surjective and injective functions. Composition of functions. The identical function on a set. Invertible functions and their inverse.

Power laws, exponentials, and logarithms in the set of real numbers. 

Cardinality of infinite sets. Countable sets. Cardinality of real numbers.

More on functions of real numbers. Domain of a function. Examples. Operations on the graph of a functions through translations, dilations, and absolute values. Examples of elementary functions. Bounded functions, monotone functions, even and odd functions, periodic functions. Trigonometric functions and their inverse.

Upper bounds and lower bounds. Bounded and unbounded subsets of the real line. Maxima and minima. Supremum and infimum. Examples.

Complex numbers. Cartesian form, real and imaginary part, conjugacy, modulus, operations with complex numbers. Representation on the complex plane. Trigonometric form. De Moivres' formulas. Roots of a complex number. Algebraic equations on complex numbers. Fundamental theorem of algebra.

Polynomials on real numbers. Algebraic and transcendental equations. Trigonometric equations, exponential equations, logarithmic equations. Exercises.

The Cartesian plane. Geometric loci on the plane: straight lines, parabolas, circles, ellipses, hyperboles. Exercises. Solution of nonlinear systems and intersection of geometric loci in the plane.

Solution of algebraic and transcendental inequalities on the real line. Exercises.

Introduction to the study of the graph of a functions of real variables. Domain, zeroes, sign. Exercises