The term normal form is used in a variety of contexts. Many of the uses in mathematics are special cases of a single situation, looked at abstractly: within an equivalence class one specifies a representative element, which is in a simplest or most manageable or otherwise tidiest and most desirable form, in terms of structure or syntax. A little more loosely, an equivalence class might contain several examples of such special, distinguished elements. For example, the Jordan normal form under similarity of matrices (link below) may mean any suitable block matrix in similarity class, and in the general case there can be several such. Mathematics is the study of quantity, structure, space and change. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... The structure of a thing is how the parts of it relate to each other, how it is put together. This contrast with process, which is how the thing works; but process requires a viable structure. ... In linguistics, syntax is the study of the rules, or patterned relations, that govern the way the words in a sentence are arranged. ... In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ...

In classical logic, propositions may be in: Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... Proposition is a term used in logic to describe the content of assertions, content which may be taken as being true or false, and which are a non-linguistic abstraction from the linguistic sentence that constitutes an assertion. ...

• Negation normal form
• Conjunctive normal form
• Disjunctive normal form
• Algebraic normal form

In formal language theory: A logical formula is in negation normal form if negation occurs only immediately above elementary propositions. ... In Boolean logic, Conjunctive Normal Form (CNF) is a method of standardizing and normalizing logical formulas. ... In Boolean logic, Disjunctive Normal Form (DNF) is a method of standardizing and normalizing logical formulas. ... In Boolean logic, Algebraic Normal Form (ANF) is a method of standardizing and normalizing logical formulas. ... In mathematics, logic and computer science, a formal language is a set of finite-length words (i. ...

• Chomsky normal form
• Greibach normal form
• Kuroda normal form

In relational database theory A formal grammar is in Chomsky normal form iff all production rules are of the form: A → BC or A → α where A, B and C are nonterminal symbols and α is a terminal symbol. ... To say that a context-free grammar is in Greibach normal form (GNF) means that all production rules are of the form: where A is a nonterminal symbol, α is a terminal symbol and X is (possibly empty) sequence of nonterminal symbols. ... A formal grammar is in Kuroda normal form iff all production rules are of the form: AB → CD or A → BC or A → B or A → α where A, B, C and D are nonterminal symbols and α is a terminal symbol. ... A relational database is a database based on the relational model. ...

• first normal form
• second normal form
• third normal form
• fourth normal form
• fifth normal form

see database normalization for all four. Database normalization (also spelt database normalisation) relates to the level of redundancy in a databases structure. ...

In linear algebra: Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...

• Jordan normal form
• Frobenius normal form

In proof theory In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K, to what extent can M be simplified into a standard... In linear algebra, the Frobenius normal form of a matrix is a normal form that reflects the structure of the minimal polynomial of a matrix. ... Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ...

• Normal form for proofs in natural deduction

In the lambda calculus An inference of natural deduction is a normal form, according to Dag Prawitz, if no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule. ... In mathematical logic, natural deduction is the name given to a class of foundational approaches for two key concepts in logic, propositions and proofs. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ...

• Beta normal form and Beta-eta normal form

In musical set theory: In the lambda calculus, a term is in beta normal form if no beta reduction is possible. ... In the lambda calculus, a term is in beta normal form if no beta reduction is possible. ... Musical set theory is a atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as sets and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation. ...

• the normal form of a pitch or pitch class set, which is the order that occupies the smallest possible span and is stacked leftmost.

To transform something into a normal form is often called normalization. In music, pitch is the perception of the frequency of a note. ... In music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... Broadly, normalization (also spelled normalisation) is any process that makes something more normal, which typically means conforming to some regularity or rule, or returning from some state of abnormality. ...

See also

• canonical form
• Smith normal form
• Normalization property