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Discrete Math Cram Sheet
- October 6,
- 1 Propositional Logic Contents
- 1.1 Truth Tables
- 1.2 Logical Equivalences
- 1.3 Rules of Inference
- 1.4 Satisfiability
- 2 Proofs
- 2.1 Well-Ordering Principle
- 2.2 Mathematical Induction
- 2.3 Strong Induction
- 3 Recurrence Relations
- 4 Number Theory
- 4.1 Divisibility
- 4.2 Primes and Factors
- 4.3 Divisors
- 4.4 Modular Arithmetic
- 5 Graph Theory
- 5.1 Notation
- 5.2 Definitions
- 5.3 Properties
- 6 Linear Algebra
- 7 Combinatorics
- 7.1 Permutations and Combinations
- 7.2 Binomial Coefficients
- 7.3 Generalized Permutations and Combinations
- 7.4 Principle of Inclusion-Exclusion
- 7.5 Derangements
- 7.6 Catalan Numbers
- 7.7 Partitions
- 7.8 Stirling Numbers
- 8 Probability
1 Propositional Logic
1.1 Truth Tables
p T T F F q T F T F F F F F F contradiction p ∨ q F F F T joint denial p 8 q F F T F converse nonimplication ¬p F F T T left negation p 9 q F T F F nonimplication ¬q F T F T right negation p ⊕ q F T T F exclusive disjunction p Z q F T T T alternative denial p ∧ q T F F F conjunction p ↔ q T F F T biconditional/equivalence q T F T F right projection p → q T F T T implication p T T F F left projection p ← q T T F T converse implication p ∨ q T T T F disjunction T T T T T tautology
1.2 Logical Equivalences
Identity p ∧ T ≡ p p ∨ F ≡ p
Domination p ∨ T ≡ T p ∧ F ≡ F
Idempotent p ∧ p ≡ p p ∨ p ≡ p
Commutative p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p
Associative p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r
Distributive p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
De Morgan’s ¬ (p ∧ q) ≡ ¬p ∨ ¬q ¬ (p ∨ q) ≡ ¬p ∧ ¬q
Absorption p ∧ (p ∨ q) ≡ p p ∨ (p ∧ q) ≡ p
Negation p ∨ ¬p ≡ T p ∧ ¬p ≡ F
Double Negation ¬ (¬p) ≡ p
Involving Biconditionals
p ↔ q ≡ (p → q) ∧ (q → p) p ↔ q ≡ ¬p ↔ ¬q p ↔ q ≡ (p ∧ q) ∨ (¬p ∧ ¬q)
¬ (p ↔ q) ≡ p ↔ ¬q
Involving Conditional Statements
p → q ≡ ¬p ∨ q p → q ≡ ¬q → ¬p p ∨ q ≡ ¬p → q p ∧ q ≡ ¬ (p → ¬q) (p → q) ∧ (p → r) ≡ p → (q ∧ r) (p → r) ∧ (q → r) ≡ (p ∨ q) → r (p → q) ∨ (p → r) ≡ p → (q ∨ r) (p → r) ∨ (q → r) ≡ (p ∧ q) → r
1.3 Rules of Inference
Modus Ponens
p → q p q
Modus Tollens
¬q p → q ¬p
Associative
(p ∨ q) ∨ r p ∨ (q ∨ r)
Commutative p ∧ q q ∧ p
Biconditional
p → q q → p p ↔ q
Exportation (p^ ∧^ q) →^ r p → (q → r)
Contraposition
p → q ¬q → ¬p
Hypothetical Syllogism
p → q q → r p → r
Material Implication p → q ¬p ∨ q
Distributive
(p ∨ q) ∧ r (p ∧ r) ∨ (q ∧ r)
Absorption p → q p → (p ∧ q)
Disjunctive Syllogism
p ∨ q ¬p q
Addition p p ∨ q
Simplification
p ∧ q p
Conjunction
p q p ∧ q
Double Negation
p ¬¬p
Disjunctive Simplification
p ∨ p p
Resolution
p ∨ q ¬p ∨ r q ∨ r
Fermat’s Little Theorem
If p is a prime number and a is a natural number, then
ap^ ≡ a (mod p)
Chinese Remainder Theorem
Let m 1 , m 2 ,... , mn be pairwise relatively prime positive in- tegers, and a 1 , a 2 ,... , an be arbitrary integers. Then the sys- tem (^)
x ≡ a 1 (mod m 1 ) x ≡ a 2 (mod m 2 ) .. . x ≡ an (mod mn)
has a unique solution modulo m = m 1 m 2 · · · mn, where x = (^) ∑nk= 1 ak Mkyk, Mk = (^) mmk , and yk is the modular inverse
of Mk modulo mk, i.e. Mkyk ≡ 1 (mod mk).
5 Graph Theory
5.1 Notation
Fundamental Notation
G graph
V vertex set
E edge set
Graph Invariants
c (G) circumference
d (u, v) distance be- tween two ver- tices
deg (v) degree of a vertex
gir (G) girth
χ (G) chromatic number
χ ′^ (G) chromatic index
δ (G) minimum de- gree
∆ (G) maximum degree
κ (G) vertex connectivity
λ (G) edge connectivity
5.2 Definitions
graph an ordered pair (V, E) where V is the set of vertices and E is the set of edges
simple a graph having neither loops nor multiple edges
multigraph a graph with multiple edges but no loops
pseudograph a graph having both loops and multiple edges
digraph a directed graph in which each edge has a direc- tion
adjacency two distinct vertices v and w in a graph are ad- jacent if the pair {v, w} is an edge
incidence a vertex v and an edge e are incident with one another if v ∈ e
degree (of a vertex v, in symbols deg (v)) the number of vertices adjacent to v
walk an alternating sequence v 0 , e 1 , v 1 ,... , ek, vk of ver- tices vi and edges ei for which ei is incident with vi− 1 and with vi for each i
path a walk whose vertices are distinct
trail a walk whose edges are distinct circuit a trail whose first and last vertices are identical
cycle a circuit where each pair of whose vertices other than the first and the last are distinct
5.3 Properties
Handshaking Lemma
In any graph the sum of the vertex degrees is equal to twice the number of edges.
v∈V
deg (v) = 2 |E|
6 Linear Algebra
7 Combinatorics
7.1 Permutations and Combinations
Permutation
A permutation or ranking of n objects is a listing of them in a certain order from first to last. The number of permutations of length k from n distinct ob- jects where repetition is not allowed is
n Pk = (n)k =^
n! (n − k)!
where (n)k denotes the falling factorial.
Combination
A combination of k objects taken from a collection of n ob- jects is simply a selection of k of those distinct objects with- out regard to order. The number of different combinations of k objects taken from a collection of n distinct objects without repetition is
nCk =
n k
n! k! (n − k)!
7.2 Binomial Coefficients
The binomial coefficient (nk) can be defined as the co- efficient of the xk^ term in the polynomial expansion of (x + 1 )n, which occurs in the binomial formula
(x + y)n^ =
n
k= 0
n k
xn−kyk
n k
n! k! (n − k)!
n − 1 k − 1
n − 1 k
n n − k
Pascal’s Triangle
Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4: 1 4 6 4 1 Row 5: 1 5 10 10 5 1 Row 6: 1 6 15 20 15 6 1 Row 7: (^1 7 21 35 35 21 7 ) Row 8: 1 8 28 56 70 56 28 8 1 Row 9: 1 9 36 84 126 126 84 36 9 1 Row 10: 1 10 45 120 210 252 210 120 45 10 1
7.3 Generalized Permutations and Combina-
tions
Permutations with Repetitions
The number of permutations of length k from n distinct ob- jects where repetition is allowed is nk.
Permutations with Duplicate Objects
The number of permutations of a multiset of n objects made up of k distinct objects can be expressed as follows: ( n n 1 , n 2 ,... , nk
n! n 1 !n 2! · · · nk!
where ni represents the multiplicity of a distinct object i in the multiset.
Combinations with Repetition (Stars and Bars)
The number of combinations of length n using k different kinds of objects is
n Rk =
n + k − 1 n − 1
n + k − 1 k
(n + k − 1 )! k! (n − 1 )!
Number of Non-negative Integer Solutions The num- ber of solutions of the equation x 1 + x 2 + · · · + xk = n in non-negative integers is (n+ k−k− 1 1 ).
Number of Positive Integer Solutions The number of solutions of the equation x 1 + x 2 + · · · + xk = n in posi- tive integers is (n k−−^11 ).
7.4 Principle of Inclusion-Exclusion
This provides an organized method/formula to find the number of elements in the union of a given group of sets, the size of each set, and the size of all possible intersections among the sets.
Two/Three Sets
Suppose that A,B, and C are finite sets. Then:
• |A ∪ B| = |A| + |B| − |A ∩ B|
• |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| −
|B ∩ C| + |A ∩ B ∩ C|
General Form
For finite sets A 1 ,... , An, one has the identity:
⋃^ n
i= 1
Ai
n
i= 1
|Ai| − ∑
1 ≤i<j≤n
∣Ai ∩ Aj
1 ≤i<j<k≤n
∣Ai ∩ Aj ∩ Ak
−... + (− 1 )n−^1 |A 1 ∩ · · · ∩ An|
=
n
k= 1
(− 1 )k+^1
1 ≤i 1 <···<ik ≤n
∣Ai 1 ∩ · · · ∩^ Aik
7.5 Derangements
A derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangements of n elements can be determined as follows:
!n = (n − 1 ) (! (n − 1 ) +! (n − 2 )) = n!
n
k= 0
(− 1 )k k!
OEIS A000166: 0, 1, 2, 9, 44, 265, 1854, 14833, 133496,
