https://teddyrookbook.com/math/discrete-math/Recent content in Discrete Math on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/discrete-math/propositions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/propositions/A statement is either true or false; "and," "or" and "not" build new statements from old ones. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/truth-tables/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/truth-tables/List every combination of truth values, 2ⁿ rows for n statements, and work out the compound statement one column at a time. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/conditional/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/conditional/P → Q is false only when P is true and Q is false; P ↔ Q is true when P and Q have the same truth value. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/logical-equivalence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/logical-equivalence/Two statements are equivalent when their truth tables match: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q, and P → Q ≡ ¬P ∨ Q. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/quantifiers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/quantifiers/"For all" (∀) and "there exists" (∃) make claims about every element of a domain, or about at least one. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/negating-quantifiers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/negating-quantifiers/¬∀x P(x) ≡ ∃x ¬P(x) and ¬∃x P(x) ≡ ∀x ¬P(x). Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/nested-quantifiers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/nested-quantifiers/The order of ∀ and ∃ matters: ∀x ∃y (y > x) is true for real numbers, but ∃y ∀x (y > x) is false. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/valid-arguments/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/valid-arguments/Modus ponens and modus tollens are valid forms of argument; affirming the consequent and denying the antecedent are not. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/counterexamples/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/counterexamples/One case where the hypothesis holds and the conclusion fails disproves a "for all" claim. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/direct-proof/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/direct-proof/Assume the hypothesis, rewrite it with definitions (an even number is 2k), and reason step by step to the conclusion. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/contrapositive-proof/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/contrapositive-proof/To prove P → Q, prove ¬Q → ¬P instead: to show "if n² is even, then n is even," show that odd n gives odd n². Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/proof-by-cases/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/proof-by-cases/Split into cases that cover every possibility (n even or odd; n = 3k, 3k + 1 or 3k + 2) and prove each one. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/contradiction-proof/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/contradiction-proof/Assume the statement is false and reason to something impossible, as in the proof that √2 is irrational. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/sets/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/sets/Describe a set by listing its elements or by a rule, and use ∈, ⊆ and |A|. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/set-operations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/set-operations/Union, intersection, difference and complement, pictured with Venn diagrams, with De Morgan's laws for sets. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/power-sets-products/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/power-sets-products/A set with n elements has 2ⁿ subsets, and A × B has |A| · |B| ordered pairs. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/functions/A function assigns each element of its domain exactly one element of its codomain; the range is the set of outputs actually reached. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/injective-surjective/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/injective-surjective/One-to-one: different inputs give different outputs. Onto: every element of the codomain is reached. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/bijections-inverses/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/bijections-inverses/A function has an inverse exactly when it is one-to-one and onto, and (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/relations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/relations/A relation on A is a set of ordered pairs; check whether it is reflexive, symmetric, antisymmetric and transitive. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/equivalence-relations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/equivalence-relations/A reflexive, symmetric and transitive relation splits its set into classes that do not overlap. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/partial-orders/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/partial-orders/Draw a partial order such as "divides" as a Hasse diagram, and list the elements in an order that respects it. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/sum-product-rules/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/sum-product-rules/Add the counts of cases that cannot happen together; multiply the counts of choices made one after another. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/complement-counting/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/complement-counting/Count "at least one" as all the possibilities minus the ones with none. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/inclusion-exclusion/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/inclusion-exclusion/|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/division-rule/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/division-rule/Divide out the orderings that look the same: MISSISSIPPI has 11!/(4! · 4! · 2!) arrangements, and n people around a round table have (n − 1)!. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/binomial-identities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/binomial-identities/Prove an identity by counting one set in two ways: C(n, k) = C(n − 1, k − 1) + C(n − 1, k), and each row of Pascal's triangle adds to 2ⁿ. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/lattice-paths/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/lattice-paths/A path of right and up steps is a string of R's and U's, so a grid m steps wide and n steps tall has C(m + n, n) shortest paths. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/stars-and-bars/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/stars-and-bars/There are C(n + k − 1, k − 1) ways to put n identical items into k boxes, and C(n − 1, k − 1) when every box gets at least one. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/pigeonhole/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/pigeonhole/If more than k · n items go into n boxes, some box gets at least k + 1; to guarantee something, plan for the worst case. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/derangements/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/derangements/Count the arrangements where nothing stays in its own place with inclusion–exclusion: D₄ = 9, and Dₙ/n! approaches 1/e. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/induction-beyond-sums/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/induction-beyond-sums/Use the case n = k to rewrite the case n = k + 1: 3 ∣ (4ⁿ − 1) for n ≥ 1, and 2ⁿ > n² for n ≥ 5. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/strong-induction/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/strong-induction/Assume every case up to k to prove case k + 1, with as many base cases as the step reaches back. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/recurrence-models/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/recurrence-models/Describe a count by how it depends on smaller cases: tilings, binary strings with no two 1s in a row, and the Tower of Hanoi. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/unrolling-recurrences/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/unrolling-recurrences/Substitute the rule into itself until a pattern appears, then check the closed form: aₙ = 2aₙ₋₁ + 1 with a₁ = 1 gives aₙ = 2ⁿ − 1. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/linear-recurrences/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/linear-recurrences/For aₙ = c₁aₙ₋₁ + c₂aₙ₋₂, solve r² = c₁r + c₂ and fit aₙ = A·r₁ⁿ + B·r₂ⁿ to the first two terms. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/repeated-root-recurrences/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/repeated-root-recurrences/When r² = c₁r + c₂ has a double root r, the solutions are aₙ = (A + Bn)rⁿ. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/big-o/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/big-o/Rank functions by how fast they grow, log n ≪ n ≪ n log n ≪ n² ≪ 2ⁿ ≪ n!; f is O(g) when f(n) ≤ C · g(n) for all large n. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/graph-basics/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/graph-basics/A graph is a set of vertices joined by edges; the degrees add up to twice the number of edges. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/special-graphs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/special-graphs/Kₙ has n(n − 1)/2 edges, K(m, n) has mn, and the cycle Cₙ has n. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/connectivity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/connectivity/A path never repeats a vertex; a graph is connected when every two vertices are joined by a path. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/adjacency-matrices/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/adjacency-matrices/Store a graph as a matrix A; the (i, j) entry of Aᵏ counts the walks of length k from i to j. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/isomorphism/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/isomorphism/Two graphs have the same structure when a relabeling of the vertices matches their edges; different degree sequences prove they do not. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/euler-circuits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/euler-circuits/A connected graph has an Euler circuit when every degree is even, and an Euler path when exactly two degrees are odd. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/hamilton-paths/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/hamilton-paths/Visit every vertex exactly once; there is no quick test, but Dirac's theorem guarantees a cycle when every degree is at least n/2. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/trees/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/trees/A tree is connected with no cycles; a tree with n vertices has n − 1 edges, and adding any edge creates a cycle. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/spanning-trees/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/spanning-trees/Kruskal's algorithm adds the cheapest remaining edge that does not close a cycle, until n − 1 edges are chosen. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/shortest-paths/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/shortest-paths/Each round, settle the closest unsettled vertex and update the distances of its neighbors. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/graph-coloring/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/graph-coloring/Color the vertices so that neighbors differ; a graph can be colored with two colors exactly when it has no odd cycle. Discrete Math · Math Ladderhttps://teddyrookbook.com/math/discrete-math/planar-graphs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/discrete-math/planar-graphs/A connected planar drawing has v − e + f = 2; since a simple planar graph has e ≤ 3v − 6, K₅ is not planar. Discrete Math · Math Ladder