Discrete Mathematics

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

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.

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.

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.

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.

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.

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 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 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.

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).

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.

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.

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.

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.

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.

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 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.

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.

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 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.

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.

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.