In other words, is a subfunction of iff whenever such that is defined, then , is defined, and .
If we set and , then , so there is no harm in assuming that and are subsets of and respectively, which we will do for the rest of the discussion.
In practice, whenever is a subfunction of , we often assume that and have the same domain and codomain. Otherwise, we would specify that is a subfunction of with domain and codomain .
For example, defined by
is a partial function, whose domain of definition is , and the partial function given by
is a subfunction of . The domain of definition of is .
Two immediate properties of a subfunction of are
the range of is a subset of the range of :
the domain of definition of is a subset of the domain of definition of :
Definition. A subfunction of is called a restriction of relative to , if , and a restriction of if .
Every partial function corresponds to a unique restriction of .
A restriction of is certainly a restriction of relative to , since , but not conversely. For example, let be the set of all non-negative integers and the ordinary subtraction. is easily seen to be a partial function. Let be the set of all positive integers. Then is a restriction of , relative to . However, is not a restriction of , for is not defined, while .
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
|Date of creation||2013-03-22 18:41:54|
|Last modified on||2013-03-22 18:41:54|
|Last modified by||CWoo (3771)|