subfunction (original) (raw)
In other words, g is a subfunction of f iff whenever x∈C such that g(x) is defined, then x∈A, f(x) is defined, and g(x)=f(x).
If we set C′=A∩C and D′=B∩D, then g⊆f∩(C′×D′), so there is no harm in assuming that C and D are subsets of A and B respectively, which we will do for the rest of the discussion.
In practice, whenever g is a subfunction of f, we often assume that g and f have the same domain and codomain
. Otherwise, we would specify that g is a subfunction of f:A→B with domain C and codomain D.
For example, f:ℝ→ℝ defined by
is a partial function, whose domain of definition is (-∞,-1]∪[1,∞), and the partial function g:ℝ→ℝ given by
is a subfunction of f. The domain of definition of g is (-∞,-1)∪(1,∞).
Two immediate properties of a subfunction g:C→D of f:A→B are
- •
the range of g is a subset of the range of f: - •
the domain of definition of g is a subset of the domain of definition of f:
Definition. A subfunction g:C→D of f:A→B is called a restriction of f relative to D, if g(C)=f(C)∩D, and a restriction of f if g(C)=f(C).
Every partial function g:C→D corresponds to a unique restriction g′:C→g(C) of g.
A restriction g:C→D of f:A→B is certainly a restriction of f relative to D, since f(C)∩D=g(C)∩D=g(C), but not conversely. For example, let A be the set of all non-negative integers and -A:A2→A the ordinary subtraction. -A is easily seen to be a partial function. Let B be the set of all positive integers. Then -B:B2→B is a restriction of -A:A2→A, relative to B. However, -B is not a restriction of -A, for n-Bn is not defined, while n-An=0∈A.