Outcome

An outcome is a distinct, possible result of an experiment. Sets of outcomes are known as events, and the set of all outcomes in a given experiment is its sample space.

If the probability distribution $p$ is known, then the probability of an outcome $s$ in the sample space $S$ occurring is: $$p(s)$$

Outcomes are typically denoted with numbers, letters, words, or symbols. For example, $4$ could represent the outcome of rolling $4$ with a die. Likewise, $H$ could represent the outcome of a coin flip resulting in heads.

Sometimes, outcomes can be collections. Consider drawing $6$ cards from a shuffled deck. Then, an outcome is a $6$-card hand, such as $\{$ K♠$,$ 5♣$,$ J♣$,$ 8♥$,$ 10♥$,$ 2♦ $\}.$ There are $\binom{52}{6}$ possible outcomes.

⚠ Here, the outcome was a set. If the order in which the cards were drawn mattered, the outcomes would instead be sequences, and there would be $P(52, 6)$ in total.

⚠ Wondering where the $52$ came from? For each of the $4$ suits $\{$ ♠$,$ ♣$,$ ♥$,$ ♦ $\}$, there are $13$ ranks $\{$ A$,$ 2$,$ 3$,$ 4$,$ 5$,$ 6$,$ 7$,$ 8$,$ 9$,$ 10$,$ J$,$ Q$,$ K $\}$ in a standard deck of cards, totalling $4 \times 13 = 52$ cards.

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