Pure Mathematics for Pre-Beginners

Describe sets explicitly by listing each element using the roster method, noting that order does not matter and elements are listed once; use membership notation X ∈ A and X ∉ A.

Describe finite and infinite sets using ellipses and the roster method, with examples like the natural numbers 1 to 100, the integers, even and odd numbers, and positive integers.

Describe sets by a shared property P and describe membership with set-builder notation, such as the set of X such that P(X), using variables and bounding sets.

Define subsets and proper subsets, illustrate subset relations and transitivity with examples from natural numbers, integers, rationals, reals, complex numbers, and visualize with Venn diagrams and a universal set.

Learn how the power set of a set comprises all subsets, illustrated with A, B, and C, including the empty set, and how its size equals two to the n.

Explore basic set operations—union, intersection, difference, and symmetric difference—through Venn diagrams, with subsets, disjointness, and key properties like commutativity, associativity, and distributivity.

Explore identities in semigroups and monoids under the binary operation star. Understand that a unique identity makes a monoid, with examples from natural numbers, integers, and power sets.

Define groups from monoids, explore identity, associativity, and inverses, and examine examples including integers, rationals, clock arithmetic, and inverse uniqueness.

Explore the field axioms, including additive and multiplicative structures, distributivity, and zero not equal to one, with examples from rational, real, complex numbers, and modular arithmetic.

Expresses every integer as A = M B + R, with 0 ≤ R < M, yielding a unique quotient and remainder.

Master gcd and lcm concepts through definitions, examples, and exercises, including greatest common divisor, least common multiple, relative primality, and computations for sets of integers.

Learn the Euclidean algorithm for computing the gcd of two positive integers, including expressing the gcd as a linear combination and working through step-by-step examples.

Explain binary relations on sets and their order properties, including reflexive, symmetric, transitive, and antisymmetric, with examples from equals, subset relation on power set, and less than in ordered sets.

Explore ordered rings and fields, showing how addition preserves order and positive elements yield positive products with Z and Q, and 1/x > 0 for x > 0.

Explore why rational numbers aren't enough, using the Pythagorean theorem on right triangles and the fact that no rational q satisfies q^2 = 2.

Define intervals of real numbers and distinguish open, closed, half-open, and infinite types. Apply interval notation with parentheses and brackets to indicate endpoints, with examples like (0,1), [0,1], and (1, infinity).

Identify closed sets in R by using complements, and verify examples like [0,1], finite unions, and arbitrary intersections as closed; distinguish half-open intervals as neither open nor closed.

Explore absolute value and distance in complex and real numbers, defined via the modulus and the Pythagorean theorem, with real and imaginary square roots.

Explore the basic topology of the complex plane by defining circles, centers, and radii. Learn about open and closed disks, neighborhoods, punctured disks, and open and closed sets.

Explore matrices over the real numbers, including 2x2 examples with zero and identity matrices, addition, and scalar multiplication, plus matrix multiplication and dimension rules.

Explore vector spaces over fields, including Q, R, and C, and verify the vector space axioms—closure, associativity, commutativity, identities, inverses, and distributivity—through examples like matrices, R^2, R^3, and F^n.