Equivalence class

An equivalence class is a subset whose elements are considered equivalent under a particular equivalence relation. Equivalence classes are "pairwise disjoint," meaning they have no elements in common with each other. The union of all equivalence classes of a set will result in that original set.

An equivalence class is denoted by $[a]$, where $a$ is any element of that equivalence class chosen to represent the entire class. Whichever representative you choose is simply a matter of preference and convenience, just stay consistent.

For an equivalence relation $\sim$ on a set $S$, the equivalence class of an element $a$ in $S$ is: $$[a] = \set{x \in S : x \sim a}$$

So then, if $a \sim b$, they must belong to the same equivalence class, meaning $[a] = [b].$ If $a \not \sim b$, $[a]$ and $[b]$ must be completely disjoint equivalence classes.

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