Probability

The probability of an event is its likelihood to occur, expressed as a real number from \(0\) (impossibility) to \(1\) (absolute certainty). Counting techniques can be applied to calculate the probability of a random event. A probability distribution associates outcomes with probabilities.

Discrete probability is the probability of an event in a sample space that is either finite or countably infinite. For a set to be countably infinite, a bijection must exist between it and \(\mathbb{Z}\). Otherwise, it's uncountably infinite.

The set of all binary strings, \(B^*\), and the set of all rational numbers, \(\mathbb{Q}\), are examples of countably infinite sets.

The sample space of throwing a dart at a target, where the outcome is the exact position of the dart on the target as an ordered pair of two real numbers, \(\mathbb{R} \times \mathbb{R} = \mathbb{R}^2\), is an example of an uncountably infinite set. If the ordered pairs consisted of integers, the sample space would have been \(\mathbb{Z} \times \mathbb{Z} = \mathbb{Z}^2\), a countably infinite set.

Logic & Proofs
Integer • Rational number • Inequality • Real number • Theorem • Proof • Statement • Proof by exhaustion • Universal generalization • Counterexample • Existence proof • Existential instantiation • Axiom • Logic • Truth • Proposition • Compound proposition • Logical operation • Logical equivalence • Tautology • Contradiction • Logic law • Predicate • Domain • Quantifier • Argument • Rule of inference • Logical proof • Direct proof • Proof by contrapositive • Irrational number • Proof by contradiction • Proof by cases • Summation • Disjunctive normal form
Set Theory
Set • Element • Empty set • Universal set • Subset • Power set • Cartesian product • String • Binary string • Empty string • Set operation • Set identity • Set proof
Functions
Function • Floor function • Ceiling function • Inverse function
Algorithms
Algorithm • Pseudocode • Command • Asymptotic notation • Time complexity • Atomic operation • Brute-force algorithm
Relations
Relation • Reflexive relation • Symmetric relation • Transitive relation • Relation composition • Equivalence relation • Equivalence class
Number Theory
Integer division • Linear combination • Division algorithm • Modular arithmetic • Prime factorization • Greatest common divisor • Least common multiple • Primality test • Factoring algorithm • Euclid's theorem • Prime number theorem • Euclidean algorithm
Induction
Proof by induction • Fibonacci sequence • Proof by strong induction • Well-ordering principle • Sequence • Factorial • Recursive definition
Combinatorics
Rule of product • Rule of sum • Bijection rule • Permutation • Combination • Complement rule • Experiment • Outcome • Sample space • Event • Probability • Probability distribution • Uniform distribution • Multiset • Sixfold way • Inclusion-exclusion principle • Pigeonhole principle
Graph Theory
Graph • Walk • Subgraph • Regular graph • Complete graph • Empty graph • Cycle graph • Hypercube graph • Bipartite graph • Component • Eulerian circuit • Eulerian trail • Hamiltonian cycle • Hamiltonian path • Tree • Huffman tree • Substring • Forest • Path graph • Star • Spanning tree • Weighted graph • Minimum spanning tree • Greedy algorithm • Prim's algorithm
Recursion
Recursion • Recursive algorithm • Correctness proof • Divide-and-conquer algorithm • Sorting algorithm • Merge sort