Axiom
Axioms are statements that you're allowed to assume are true to aid in the formulation of proofs. Depending on your audience, certain axioms may be so obvious that you don't need to mention them when they are used in a step of your proof. However, when a step is complicated, mention exactly which axiom or theorem you're using.
Here's a list of some typical axioms:
- The rules of algebra always apply. (If you do weird stuff, clarify it!)
- The set of integers is closed under addition, subtraction, and multiplication. (This means sums, differences, and products of integers are always integers!)
- All integers are either even or odd. (True because of the Division Algorithm!)
- If \(n\) is an integer, there is no integer between \(n\) and \(n+1.\)
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
Algorithms
Relations
Number Theory
Induction
Combinatorics
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