Pure Mathematics for Beginners

Describe how a set is determined by its elements, not order, using epsilon for membership, with finite and infinite sets, ellipses, set-builder notation, and cardinality, including the empty set.

Explore subset relations: A is a subset of B if every element of A is in B, using for all x, universal set, empty set, Venn diagrams, and power sets.

Explore binary operations and closure by examining how multiplication on {0,1} stays in the set while addition does not, and formalize the concept with function notation.

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.

Discover monoids and identity within semigroups by analyzing operations such as addition and multiplication, and note how zero or one serves as the identity, with counterexamples using min and projection.

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

Evenness: A=2B with B integer; sum of evens is even. divisibility: A=KB with B integer; K divides A; product of multiples of K is a multiple of K.

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.

Learn the field axioms: a set F with addition forms a commutative group; nonzero elements form a commutative group under multiplication; distributivity holds; zero ≠ one. Integers form a ring.

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.

Show why rational numbers cannot square to two using a Pythagorean proof and a contradiction, then introduce real numbers as the ordered field that contains a squared equal to two.

Defines the completeness property of an ordered field, explains upper and lower bounds, supremum and infimum, with examples over rationals and real numbers, and proves Archimedean and density theorems.

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.

Examine how the real numbers limit solving polynomial equations and see that complex numbers form an algebraically closed field where all polynomial equations can be solved.

Plot complex numbers as points in the complex plane, identify real and imaginary parts, and learn addition, multiplication, and division via conjugates in the complex field.

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.

Learn how a subset becomes a subspace by containing the zero vector and remaining closed under addition and scalar multiplication, with examples from R^2, R^3, and polynomials.

define linear combinations and spans in vector spaces, show how a basis provides a minimal spanning, independent set, and explain linear independence, dependence, and when the span equals the space.