Discrete Mathematics

Define statements as sentences that are either true or false, illustrated by 2+2=4 (true) and 2+2=5 (false); show non statements like 'Fridays are nice' and use P to represent them.

Explore truth tables to determine when compound statements are true or false. Compare conjunction, inclusive and exclusive or, and negation using P and Q.

Assess logical equivalence by constructing truth tables and comparing truth values for all substitutions, using and, not, and double negation examples.

Explore tautologies and contradictions through truth tables, using examples with P and not P to show a statement that is always true or always false.

Apply De Morgan's laws in logic to negate P and Q as not P or not Q, and negate P or Q as not P and not Q.

Explore logical equivalence laws, including commutative, associative, and distributive laws, and learn via truth tables, tautology and contradiction concepts, with practice proving these laws using a downloadable reference.

Explore how biconditional statements express both directions using if and only if, and examine their truth conditions with P and Q through study and pass examples.

Explore how logic shapes computing by examining circuits, series and parallel switches, light bulbs as examples, the truth tables for and and or, and binary digits.

Learn how black boxes model circuits by mapping inputs to outputs, and how not gates, and gates, and or gates work. Explore a tool to build and test simple circuits.

Explore boolean expressions by combining boolean variables with not, and, or; translate circuits into expressions and back.

Turn truth tables into circuits by identifying true-output rows, forming and not expressions, and combining them with or to create a circuit using and gates, not gates, and or gates.

Define quantified statements and the universal quantifier, expressing that all elements of a set meet a condition, with real-number examples and counterexamples that disprove universal statements.

Learn to negate universal and existential quantified statements using counterexamples and universal negations, with examples like all men wear hats and set members.

Practice end-of-section extra problems by downloading the PDF with solutions to verify your steps and solidify the concepts we've studied, and ask questions anytime.

Discover the properties of numbers, including even and odd numbers, primes, and composites, as you begin an introduction to number theory.

Explore parity in integers by defining even numbers as 2k and odd numbers as 2k+1, and learn how addition and multiplication preserve parity with concrete examples.

Explore divisibility of integers, including the formal definition n = d·k and rules for 1, 2, 3, 4, and 5, noting A|B and B|A need not be equal.

Define prime numbers as integers greater than one with only one and themselves as positive divisors. Explore the fundamental theorem of arithmetic, primes versus composites, and primality tools.

Explore prime factorization by breaking numbers into prime factors, using the smallest primes to divide until one, and learn the unique prime factorization theorem.

Download the pdf of extra practice problems with solutions to check your steps and reinforce the concepts we studied, and practice as much as possible.

Explore core terminologies in discrete mathematics, including conjectures, theorems, axioms, lemmas, and corollaries, and how proofs turn conjectures into proven statements.

Explore proofs by contrapositive, an indirect method where not q implies not p, leveraging logical equivalence to prove statements when direct proofs fail, with examples.

Explore proofs by contradiction, learning to assume the opposite and derive a contradiction, as shown with infinite primes and the irrationality of sqrt(2).

Explore proofs by exhaustion, or proofs by cases, which divide a statement into finite cases and prove each to establish the overall result.

Explore existence and uniqueness proofs, distinguishing constructive from nonconstructive methods with prime number examples and counterexamples. Learn how to verify existence and prove uniqueness in equations.

Explore proofs by induction in discrete mathematics, using the dominoes example to illustrate base cases, inductive steps, and the inductive hypothesis, culminating in the sum formula n(n+1)/2.

The lecture on induction demonstrates proving statements by induction, detailing base case, inductive hypothesis, and proving k+1, with multiple examples of algebraic identities and divisibility claims.

Download the extra practice problems pdf to practice the topics and concepts studied so far, use the included solutions to check your steps, and ask questions if needed.

Explore the fundamentals of functions, including evaluation, domain and range, function composition, and even and odd functions, to build a solid foundation in discrete mathematics.

Define a function as a relation that maps each input in the domain to exactly one output in the range, with concepts like domain, range, and f(x)=x^2.

Learn how to evaluate a function by substituting a value for the placeholder X, applying the rule, and understanding domain and range with independent and dependent variables.

Learn how a function's domain is the set of inputs that keep it defined, with examples like x+1 (all real numbers), 1/(1−x) (excluding 1), and 1/(x^2−x−2) (excluding 2 and −1).

Learn how to represent a function as a graph by pairing inputs with outputs and plotting points, then connect them to visualize functions like x+1 and x^2, noting domain, range.

Explore the Dismas graphing calculator to graph functions, edit live, and explore derivatives and integrals using online and offline modes across Chrome, iOS, and Android.

Identify a function's domain and range from its graph by listing all X values (domain) and all Y values (range); note real-number domains and possible gaps in piecewise graphs.

Explore function composition by placing one function inside another to form a new function, using notation like F∘G and evaluating F(G(x)) or G(F(x)) with examples such as F(x)=x^2.

Determine if a function is even or odd by evaluating f(-x) and comparing to f(x) or -f(x). Use examples like x^2 and x^3, and note even powers don't guarantee evenness.

Study one-to-one (injective) functions by ensuring each input maps to a unique output, and apply the horizontal line test to distinguish injective from non-injective cases.

Explore onto (surjective) functions by contrasting them with injective mappings, define domain, code domain, and range, and show that a surjective function maps every code-domain element to some domain element.

Explore inverse functions and one-to-one mappings, learn how domains and ranges switch under inversion, and apply algebraic steps to find F^{-1} from examples like 3x-2 and (2x+3)/(x-1).

Master polynomial long division by using the highest-degree term, subtracting, and bringing down terms to reveal a quotient and remainder, with dividend = divisor times quotient plus remainder.

Download the attached PDF for extra practice problems on the topics studied so far, with solutions to verify your steps; practice to solidify concepts and ask questions.

Explore relations on sets, including the inverse, and analyze properties such as reflexivity, symmetry, and transitivity, leading to equivalence relations and classes.

Discover how discrete mathematics defines relations among objects, using examples with sets and numbers, including rules like X<Y and X is a factor of Y, to form related pairs.

Define relations on sets as subsets of A×B with a domain A and codomain B. See examples: a less-than rule, a real-number relation x^2+y^2=1, and a cardinality-based subset relation.

Identify the inverse of a relation by swapping the coordinates of ordered pairs from A to B, yielding the inverse relation from B to A.

Explore reflexivity, symmetry, and transitivity in relations in discrete mathematics. Analyze a set with x minus y divisible by three and identify the related ordered pairs.

Explore reflexivity, symmetry, and transitivity with concrete examples on a set and relation, evaluating reflexivity and symmetry and identifying transitivity across cases.

Discover how equivalence classes are formed by an equivalence relation R on a set A, where the class of a is the set of x in A related to a.

Introduce graph theory by defining graphs and subgraphs, vertex degree, adjacency, and incidence. Explore isomorphism, and the notions of walks, trails, and paths as foundational topics.

Explore how graphs represent relationships through handshakes among five people, defining vertices and edges; a graph consists of vertex set V and an edge set E with endpoints in V.

Determine the degree of a vertex as the number of edges with it as an endpoint, counting loops as two and isolated vertices, and form degree sequence and graph degree.

Explore adjacency and incidence in graphs by examining how vertices connect via edges, how edges share a vertex, how loops affect adjacency, and how edges are incident on endpoints.

Represent graphs efficiently by building an adjacency matrix that records the number of edges between each pair of vertices, including loops, with size equal to the vertex count.

Learn how isomorphism identifies graphs with the same number of vertices and edges and identical edge endpoints, even with relabeled vertices.

Explore walks in graphs, distinguish trails from paths, and learn how distance and simple circuits measure shortest routes and loop structures.

Explore how eccentricity, diameter, and radius relate to distance in graphs by computing each vertex's farthest distance and identifying central and peripheral vertices.

Explore connectedness in graphs by examining walks between vertices, components, and how removing edges or vertices creates disconnecting or separating sets, bridges, and measures like edge connectivity and vertex connectivity.

Explore how the seven bridges of Königsberg map to graphs, then use vertex degree parity to determine when Euler trails or Euler circuits exist in connected graphs.

Use Flurry's algorithm to find Euler circuits and trails by starting at a vertex, deleting edges in a replica, and avoiding any edge whose removal disconnects the graph.

Explore Hamiltonian paths and circuits, where each vertex is visited exactly once and circuits return to the start. Compare to Euler trails and the absence of an existence criterion.

Apply Ore's theorem to identify hamiltonian graphs by checking that a simple graph with at least three vertices has non-adjacent vertices whose degree sum meets or exceeds the vertex count.

Download the extra pdf with practice problems and solutions to check your work and reinforce the topics studied. Practice as much as you can to ensure the concepts settle in.

Explore fundamental statistics concepts in this introductory lecture, covering mean, median, mode, range, and outlier variants, and previewing other engaging topics.

Define statistics as the science of planning studies and drawing conclusions from data. Explain data as observations; population is all elements, and a sample is a portion.

Explain arithmetic mean as the center of a dataset by dividing the sum of values by their count, illustrated with the dataset 15, 16, 20, 22, 30.

Discover how median, a measure of center, uses sorted data; pick the middle value for odd counts or average the two middle values for even counts.

Define an outlier as a data value outside the general pattern, and show how one outlier can distort the mean and the range.

Explain standard deviation as a measure of data dispersion, show its relationship to variance, and demonstrate calculation for population and sample using a dataset and a calculator.

Practice makes mastery in discrete mathematics with an attached PDF of extra problems and solutions; download, solve, and verify your steps to solidify concepts, and ask questions if needed.

Define n! as the product of all positive integers up to n, with 0! = 1, and illustrate simplifications such as 16!/14! = 16×15 and (n+1)!/(n-1)! = n^2+n.

Explore the fundamental counting principle, also called the basic counting principle, using independent cases and the multiplication rule, and handling dependency with the addition principle through practical examples.

Explore how order matters in permutations, contrasting with salads where order is irrelevant, and apply repetition and non-repetition cases using factorials and the notation P(n,r).

Explore the difference between permutations and combinations, including when repetition is allowed or not, and apply the n choose r and related formulas with donuts and lottery examples.

Discover the pigeonhole principle: when more items than containers, at least one container holds multiple items. Apply it to counting, blue and green gloves, and the birthday problem.

download the extra practice problems pdf and work through the exercises to reinforce the concepts studied, using the included solutions to verify your steps.

Explore the fundamentals of sequences and series, including arithmetic and geometric sequences, and learn about partial sums.

Identify arithmetic sequences by recognizing a fixed common difference and using the formula a_n = a + (n-1)d to find terms, illustrated with examples and a thousandth term calculation.

Dissect geometric sequences by multiplying by a fixed common ratio, identify the first term, and apply the nth term formula a r^(n-1) to find any term.

Learn to find partial sums of arithmetic sequences using the formulas S_n = n/2 [2a + (n-1)d] or S_n = n/2 (a_1 + a_n), and apply sigma notation to sums.

Explore series by revisiting sequences, including arithmetic and geometric progressions, and learn how partial sums lead to convergence or divergence in infinite series.