第五章
Relations
and Functions
5.1
Cartesian Products and Relations
For sets A,B the Cartesian product, or cross product, of A and B is denoted by AxB and equals{(a,b)|a ε A,b ε B}.
For
sets A,B, any subset of AxB is called a (binary) relation from A to
B. Any subset of AxA is called a (binary) relation on A.
For
nonempty sets A, B, a function, or mapping, from A to B, denoted
f
:A → B, is a relation from A to B in which every element of A
appears exactly once as the first component of an ordered pair in the
relation.
For
the function of f:A → B, A is called the domain(定義域)
of f and B the codomain(對應域)
of f. The subset of B consisting of those elements that appear as
second component in the ordered pairs of f is called the range(值域)
of f and is also denoted by f(A) because it is the set of images
under f.
A
domain f:A → B is called one-to-one, or injective, if each element
of B appears at most once as the image of an element of A.
A
function of f:A → B is called onto, or surjective, if f(A)=B—that
is, if for all b ε B there is at least one a ε A with f(a)=b.
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