Discrete Mathematics: Beginner's Complete, Math Crash Course

Learn how direct proofs establish conclusions from hypotheses by constructing verifications, with examples on divisibility and even/odd integers, and explore core math terms like axioms, definitions, theorems, lemmas, and corollaries.

Learn how indirect proofs use contradiction and contrapositive by assuming the opposite and deriving a contradiction, with examples like even implies odd.

Explore induction as a three-step proof method—base case, induction hypothesis, and induction step—demonstrated with example proofs that verify equations via substitution and algebra.

Explore trivial proofs, proofs by cases, and circular-argument proofs in discrete mathematics, illustrating how propositions and equations establish equivalence and multi-step reasoning.

Practice core proof techniques in discrete mathematics through exercises on even sums, equivalence of statements, contradiction, induction, and divisibility by seven.

Define sets by elements and membership, and show table, descriptive, and set-builder forms. Explain empty set, subset and proper subset, equality, and illustrate with Venn diagrams showing membership and non-membership.

Discover natural numbers, integers, rational numbers, and real numbers, and learn how natural numbers ⊆ integers ⊆ rational numbers ⊆ real numbers, with examples of subset relations and decimal representations.

Explore complement sets by identifying the universal set u, the absolute complement A^c, and the relative complement B \ A, with real-number examples.

Explore the Cartesian product of sets and how ordered tuples form by pairing elements from A and B. Emphasizes that order matters and shows enumeration with examples.

Explore core set operations—union, intersection, and symmetric difference—through visual Venn diagrams, and prove subset relations and universal set properties with complements.

Explore De Morgan's laws in set theory: the complement of a union equals the intersection of complements, and the complement of an intersection equals the union of complements, with examples.

Explore the foundations of Boolean algebra, including the set operations and, or, and not, and axioms like closure, identity, inverse, distributive, and absorption, with practical proofs.

Apply set theory fundamentals through guided exercises: solve quadratic equations in natural numbers, build cartesian products, enumerate subsets, and prove relations with subset and Boolean algebra rules.

Explore ordered pairs, where order matters in (a, b) with a in A and b in B, and solve (x+2y, 1) = (3, x-2y) to find x and y.

Learn how domain and range describe a relation: domain is the first elements, range the second. In A = {1,2,3,4}, B = {3,4,6,7}, domain is {1,2,3,4} and range {3,4,6,7}.

Explore relations and functions, defining a function as a relation where every x in the domain has a unique image in the range, with domain and range examples.

Explore inverse relations by swapping domain and range, forming the inverse as ordered pairs from the original relation, and for functions swap x and y to get f^{-1}(x)=(x-1)/2.

Define composition of relations between sets and illustrate with examples, then analyze when a relation is reflexive, symmetric, and transitive to determine equivalence.

Discover injective and surjective functions, using domain A to B, check one-to-one correspondences, and apply cardinality to determine bijectivity.

Explore exercises on functions and relations, solving domain and range, finding inverse relations, testing reflexivity, symmetry, and transitivity, and analyzing function composition and surjectivity.

Learn the basics of graphs: vertices and edges, loops, isolated vertices, and directed versus undirected graphs; understand how vertex degree counts incident edges, with loops contributing two.

Identify simple graphs with no loops or parallel edges and complete graphs K_n where every pair of vertices connects. Use the edge count n(n-1)/2.

Explore bipartite graphs, where vertices split into two disjoint sets with edges only between sets; learn to recognize complete bipartite graphs that connect every vertex across sets.

Explore paths in graph theory by defining a path as a sequence of non-repeated edges between vertices, noting path length, and distinguishing simple paths with no repeated vertices through examples.

Identify circuits in graphs as closed paths that start and end at the same vertex, with simple circuits having no repeated vertices; note acyclic graphs contain no circuits.

discover Euler circuit and Euler path; circuits use all edges and return to the same vertex, while paths use all edges but do not close, determined by vertex degrees.

Explore hamiltonian paths and hamiltonian circuits in graph theory, defining a path that visits every vertex once and a circuit that returns to the start, with example graphs and solutions.

Explore trees as a graph where any two vertices connect by exactly one path, with no loops or parallel edges, leaves have degree one, and edges equal n minus one.

Explore tree types, define rooted trees, height, and relationships such as parent, child, siblings, and ancestors; determine binary and full binary trees from examples.

Explore basic graph concepts in discrete math by examining vertex degrees, edges, circuits, and hamiltonian circuits, plus identifying tree properties across four example graphs.

Explore the division algorithm, expressing x as yq + r with 0 ≤ r < y, and divisibility theorems like x divides y and x divides y ± z.

Explore exercises on divisibility, proving that x divides a y plus b for integers a and b, and that x divides x y, with cube difference and three consecutive numbers.

Explore gcd via the Euclidean algorithm and perform divisions with remainders. Derive Bezout coefficients for integers.

This lesson in discrete mathematics teaches the Euclidean algorithm through gcd computations, including gcd(168, 30) via division with remainders and deriving Bezout coefficients with the extended Euclid using numbered exercises.

Apply the fundamental principle of counting, using the product of independent choices to count outcomes, illustrated by coin tosses and a three-letter, three-digit puzzle with no repetition.

Apply the rule of product and the fundamental principle of counting to solve exercises on combinations and permutations, including tournament winners, seating couples, purchases, and non-repeating three-digit numbers.

Explore permutations using factorials and the nPk formula to count rearrangements. Apply to examples like seven distinct numbers, three-digit codes with unique digits 0–9, and five-digit numbers with repeated digits.

Learn the difference between combinations and permutations, where order matters or does not, using 7 choose 3 and factorial formulas to count selections.

Master permutations and combinations through hands-on exercises, including group selection, rearranging letters in mathematics, forming three-digit numbers from digits 1–3, and three-letter words from love.