In Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that... mathematics, the notion of restriction finds a general definition in the context of In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... sheaves.

Often, the following definition will be sufficient:

If f: E -> F is a (partial) In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). The concept of a function is fundamental to virtually every branch of mathematics... function from E to F, and A is a A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y... subset of E, then the restriction of f to A is the (partial) function

having the graph .

(In rough words, it is "the same function", but only defined on .)

More generally, the restriction of a In mathematics, the concept of binary relation is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. But also functions are a special case of binary... binary relation is usually defined in the same way. (One could also define a restriction to a subset of E x F, and the same applies to n-ary In mathematics, a relation is a generalization of arithmetic relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation and relational algebra. A relational database stores data in relations called tables. Their data manipulation languages provide operations... relations. These cases do not fit into the scheme of In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... sheaves.)

Examples

1. The restriction of the non In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. (This is in contrast to a many-to-one function, which may map two distinct input values to the same output value.) Note that the phrase... injective function to is the injection .
2. The canonical injection of a set A into a A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y... superset E of A.