Domain

The domain of a variable is the set of all possible values it can attain. Often, domains will be implied. For example, when talking about whether a variable is even and odd, it's only natural to assume that the domain of that variable is the set of integers. However, sometimes it's not so obvious what the domain of a variable is. That's why it's best to define exactly what set of values is the domain for a particular variable, because the meaning of a statement will change under different domains. Consider the following statement:

There exists an \(x\) such that \(x^2=2.\)

But what sort of number is \(x\)? Is it a real number? Or an integer? The statement's truth value depends on the domain you select:

There exists a real number \(x\) such that \(x^2=2.\)

If \(x=\sqrt 2\), then the above statement is true. However, what if the domain of \(x\) was the set of integers?

There exists an integer \(x\) such that \(x^2=2.\)

No integer exists that is the square root of 2. Therefore, the above statement is false.

So, domains matter. Make sure to specify them for your variables!

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