7.1 Sets and Mappings
387
Let X and Y be two sets, the set composed of all the ordered pairs consisting of
the element x of X and the element y of Y is called the Cartesian product, direct
product, produce set or product set of X and Y , it is denoted by X × Y , that is
X × Y = {( x, y)|x ∈ X, y ∈ Y }
where x and y are called the projection of the ordered pair (x, y) on X and Y
respectively. When one of X and Y is an empty set, it is stipulated that X × Y = Φ.
The product of n X is usually represented by X
n .
Let X 1 , X 2 , …, X n (n 2) be n sets, then their Cartesian product is defined as
X 1 × X 2 × · · · × X n = {(x 1 , x 2 , · · · , x n )|x i ∈ X i , i = 1, 2, · · · , n}
where each X i (i = 1, 2, · · · , n) is called the coordinate set of the Cartesian product.
For example, the Cartesian product of two real straight lines R s is the real plane
R
2 , R
n is the Cartesian product of the n real straight lines R s.
The set of the ordered n-tuple (x 1 , x 2 , · · · , x n ) for the real numbers x 1 , x 2 , …, x n
is called the Cartesian space, it is denoted by R
n .
Let X and Y be two nonempty sets, if according to certain rule f , such that for
each element x in the set X , in the set Y there corresponds unique determined element
y, then f is called a map or mapping from X to Y , it is denoted by f : X → Y or
f : x → y(x ∈ X ). A mapping is sometimes called the operator, transformation
or function. Where, y is called the image of x under the mapping f , it is denoted
by f (x), that is y = f (x), sometimes it is also written as y = f x or f : x → y, the
meaning is that f maps x into y. x is called the preimage of y under the mapping
f . The set X is called the domain of definition or domain of the mapping f , it is
denoted by D f or D( f ), that is D f = D( f ) = X . The set composed of the images
of all the elements in set X is called the range or value field of the mapping f , it
is denoted by R f , R( f ) or f (X ), that is R f = R( f ) = f (X ) = { f (x)|x ∈ X }.
Note that for each element x in X , although there is the corresponding element y
in Y , namely the image y of x is unique, the elements in X and the element in Y
are not necessarily a one-to-one correspondence, namely the preimage x of y is not
necessarily unique. In general, the range f (X ) of the mapping f is only a subset of
Y , that is R( f ) ⊂ Y , and that is not necessarily all Y .
Let f : X → Y , and f (X ) = Y , namely each element in Y is the image of an
element in X , as much as to say the images of the elements in X full of all Y , then f
is called the surjection, surjective, surmorphism or epimorphism from X to Y .
Let f : X → Y , if for arbitrary x 1 , x 2 ∈ X , when x 1 = x 2 , there is f (x 1 ) = f (x 2 ),
namely to every y ∈ f (X ), there corresponds unique element x in X , the f is called
the injection or injective mapping from X to Y . The injection is also called the
invertible mapping or one-to-one mapping.
If f : X → Y is both a surjection and an injection, then f is called the bijection,
bijective mapping or correlation, it is also called the one-one correspondence or
one-one mapping of X and Y .
387
Let X and Y be two sets, the set composed of all the ordered pairs consisting of
the element x of X and the element y of Y is called the Cartesian product, direct
product, produce set or product set of X and Y , it is denoted by X × Y , that is
X × Y = {( x, y)|x ∈ X, y ∈ Y }
where x and y are called the projection of the ordered pair (x, y) on X and Y
respectively. When one of X and Y is an empty set, it is stipulated that X × Y = Φ.
The product of n X is usually represented by X
n .
Let X 1 , X 2 , …, X n (n 2) be n sets, then their Cartesian product is defined as
X 1 × X 2 × · · · × X n = {(x 1 , x 2 , · · · , x n )|x i ∈ X i , i = 1, 2, · · · , n}
where each X i (i = 1, 2, · · · , n) is called the coordinate set of the Cartesian product.
For example, the Cartesian product of two real straight lines R s is the real plane
R
2 , R
n is the Cartesian product of the n real straight lines R s.
The set of the ordered n-tuple (x 1 , x 2 , · · · , x n ) for the real numbers x 1 , x 2 , …, x n
is called the Cartesian space, it is denoted by R
n .
Let X and Y be two nonempty sets, if according to certain rule f , such that for
each element x in the set X , in the set Y there corresponds unique determined element
y, then f is called a map or mapping from X to Y , it is denoted by f : X → Y or
f : x → y(x ∈ X ). A mapping is sometimes called the operator, transformation
or function. Where, y is called the image of x under the mapping f , it is denoted
by f (x), that is y = f (x), sometimes it is also written as y = f x or f : x → y, the
meaning is that f maps x into y. x is called the preimage of y under the mapping
f . The set X is called the domain of definition or domain of the mapping f , it is
denoted by D f or D( f ), that is D f = D( f ) = X . The set composed of the images
of all the elements in set X is called the range or value field of the mapping f , it
is denoted by R f , R( f ) or f (X ), that is R f = R( f ) = f (X ) = { f (x)|x ∈ X }.
Note that for each element x in X , although there is the corresponding element y
in Y , namely the image y of x is unique, the elements in X and the element in Y
are not necessarily a one-to-one correspondence, namely the preimage x of y is not
necessarily unique. In general, the range f (X ) of the mapping f is only a subset of
Y , that is R( f ) ⊂ Y , and that is not necessarily all Y .
Let f : X → Y , and f (X ) = Y , namely each element in Y is the image of an
element in X , as much as to say the images of the elements in X full of all Y , then f
is called the surjection, surjective, surmorphism or epimorphism from X to Y .
Let f : X → Y , if for arbitrary x 1 , x 2 ∈ X , when x 1 = x 2 , there is f (x 1 ) = f (x 2 ),
namely to every y ∈ f (X ), there corresponds unique element x in X , the f is called
the injection or injective mapping from X to Y . The injection is also called the
invertible mapping or one-to-one mapping.
If f : X → Y is both a surjection and an injection, then f is called the bijection,
bijective mapping or correlation, it is also called the one-one correspondence or
one-one mapping of X and Y .