Pure Mathematics for Beginners

Explore semigroups and associativity through binary operations on sets, with concrete examples like natural numbers and integers under addition or multiplication, and a counterexample with subtraction.

Examine how the integers with addition and multiplication form a ring, and master the ring axioms, distributivity, and the roles of zero and one.

Explore the principle of mathematical induction and the well ordering principle, establishing base case and inductive step to prove statements about natural numbers, with examples on evenness and inequalities.

Delve into ordered rings and fields, the role of positive elements, and how natural numbers embed inside any ordered field, with examples from integers and rationals.

Intervals are sets of real numbers containing every point between endpoints, including open, closed, half-open, and unbounded forms on the real line, with bounded ones matching nine forms.

Explore operations on sets, including union, intersection, difference, and symmetric difference, with Venn diagrams and interval examples, plus properties like associativity, commutativity, and distributivity.

Explore open and closed sets in the real numbers, using intervals and complements to understand unions, intersections, and boundary behavior.

Explore absolute value and distance for real or complex numbers, including square roots, the modulus, and the triangle inequality via the Pythagorean framework.

Defines basic topology in the complex plane, including circles, centers, radii, and disks. Describes open and closed disks, neighborhoods, punctured disks, deleted neighborhoods, and interval criteria for open sets.

Explore vector spaces over fields, verify the six vector space axioms, and study examples including complex numbers, R^2, R^3, matrices, and polynomials of degree at most two.