Definitions

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A definition can be

• elementary (without reference to set theory)
• recursive (or inductive) (in terms of itself, with base cases to avoid infinite circularity)
• (...)

Concept/term definition

In mathematics, a definition is used to give a precise meaning to a new [mathematical] concept or term, i.e. a non-primitive notion. Undefined concepts/terms (primitive notions) and unproved statements/axioms (primitive statements) are the foundations on which all, i.e. definitions and theorems, of mathematics is constructed.

Notation definition

Well-defined notation

A notation that is unambiguous is said to be well-defined.

For real numbers (complex numbers and quaternions), the product

a  ×  b  ×  c

is unambiguous because

(a b)  c = a  (b c)

, multiplication over the real numbers (complex numbers and quaternions) being associative. Thus the notation is well-defined.

Ill-defined notation

A notation that is ambiguous is said to be ill-defined, i.e. not well-defined.

For octonions, the product

a  ×  b  ×  c

is ambiguous because

(a b)  c   ≠   a  (b c)

, multiplication over the octonions not being associative. Thus, without the parentheses, the notation is ill-defined.

Set definition

Intensional definition

An intensional definition, also called a connotative definition, of a set specifies its intension, which are the necessary and sufficient conditions for an element to be a member of that set. Any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition.

Obviously, an infinite set can only be defined by intension!

Extensional definition

An extensional definition, also called a denotative definition, of a set specifies its extension, which is a list naming every element being a member of that set.

Obviously, an infinite set cannot be defined by extension!

Function definition

Well-defined function

All functions are well-defined binary relations: given two ordered pairs

(a, b)

and

(c, d  )

, the function

f

is well-defined iff whenever

a = c

(terms from the domain) it is the case that

b = d

(images in the codomain). The contrapositive of this statement, which is equivalent, says that

b   ≠   d

implies

a   ≠   c

.

Ill-defined function

A binary relation is said to be ill-defined, i.e. not well-defined, if some term of the domain has multiple or ambiguous values (images) in the codomain.

Undefined function

A function that is not well-defined is not the same as a function that is undefined. For example, if

f  (x) = 1 / x

, then

f  (0)

is undefined, but this has nothing to do with the question of whether

f  (x) = 1 / x

is well-defined. It is; 0 is simply not in the domain of the function.