F u n c t i o n s
函数
DEFINITION 1,
1.2.2 函数
Functions
/Mapping 映射
F u n c t i o n s
函数
DEFINITION 1,
Let A and B be sets,A function f from A to B is an
assignment of exactly one element of B to each element
of A,We write f(a) = b if b is the unique element of B
assigned by the function f to the element a of A,If f is a
function from A to B,we write f,A→B,
F u n c t i o n s
函数
DEFINITION 2,
If f is a function from A to B,we say that A is the
domain of f and B is the codomain of f,If f (a) = b,we say
that b is the image of a and a is a pre-image of b,The
range of f is the set of all images of elements of A,Also,if f
is a function from A to B,we say that f maps A to B.
A：定义域 /domain of f a,b 的原象 /pre-image
B：陪域 /codomain of f b,a的象 /image
f（ A）：值域 /range of f
F u n c t i o n s
函数
DEFINITION 3,
Let f1 and f2 be function from A to R,Then f1 + f2
and f1 f2 are also functions from A to R defined by
(f1 + f2 )(x) = f1(x) + f2(x),
(f1 f2)(x) = f1(x) f2(x).
当 B=Real Set
F u n c t i o n s
函数
DEFINITION 4,
Let f be a function from the set A to the set B and let
S be a subset of A,The image of S is the subset of B that
consists of the images of the elements of S,We denote
the image of S by f (S),so that
f (S) = { f (s)∣ s∈ S}.
A的子集 S的象
F u n c t i o n s
函数
DEFINITION 11,
Let f be a function from the set A to the set B,The
graph of the function f is the set of ordered pairs {(a,
b)∣ a∈ A and f (a) = b}.
F u n c t i o n s
函数
EXAMPLE 20
Display the graph of the function f(x) = x2 from the
set of integers to the set of integers.
F u n c t i o n s
函数
DEFINITION 1,
函数的分类
F u n c t i o n s
函数
DEFINITION 5,
A function f is said to be one-to-one,or injective,if
and only f (x) = f (y) implies that x = y for all x and y in
the domain of f,A function is said to be an injection if it
is one-to-one.
一对一，单函数，单射
F u n c t i o n s
函数
EXAMPLE 6
Determine whether the function f
from {a,b,c,d} to (1,2,3,4,5) with
f(a) = 4,f(b) = 5,f(c) = 1,and f(d) = 3
is one-to-one.
F u n c t i o n s
函数
DEFINITION 6,
A function f whose domain and codomain are
subsets of the set of real numbers is called strictly
increasing if f(x) < f(y) whenever x < y and x and y are
in the domain of f,Similarly,f is called strictly
decreasing if f (x) > f (y) whenever x < y and x and y are
in the domain of f.
F u n c t i o n s
函数
DEFINITION 7,
A function f from A to B is called onto,or surjective,
if and only if for every element b∈ B there is an element
a∈ A with f (a) = b,A function f is called a surjection if
it is onto.
映上的，满函数，满射
F u n c t i o n s
函数
EXAMPLE 9
Let f be the function
from {a,b,c,d} to {1,2,3}
defined by f(a) = 3,f(b) = 2,f(c) = 1,and f(d) = 3.
Is f an onto function?
F u n c t i o n s
函数
DEFINITION 8,
The function f is a one-one correspondence,or a
bijection,if it is both one-to-one and onto.
一一对应，双射
F u n c t i o n s
函数
DEFINITION 9,
Let f be a one-to-one correspondence f from the set A
to the set B,The inverse function of f is the function that
assigns to an element b belonging to B the unique
element a in A such that f (a) = b,The inverse function
of f is denoted by f –1,Hence,f –1(b) = a when f(a)=b.
逆函数
F u n c t i o n s
函数定理 1 设 f:A→B 且是 双射,则
f –1,B → A 也是双射
F u n c t i o n s
函数
DEFINITION 10,
Let g be a function from the set A to the set B and let
f be a function from the set B to the set C,The
composition of the functions f and g,denoted by f o g,is
defined by
(f o g)(a) = f (g (a)).
函数的复合运算，积运算
F u n c t i o n s
函数
EXAMPLE 18
Let f and g be the functions from the set of integers to
the set of integers defined by f(x) = 2x + 3 and g(x) = 3x
+ 2,What is the composition of f and g? What is the
composition of g and f?
(f o g )(x) = f(g(x)) = f(3x + 2) = 2(3x + 2) + 3 = 6x + 7
(g o f )(x) = g(f(x)) = g(2x + 3) = 3(2x + 3) + 2 = 6x + 11
F u n c t i o n s
函数定理 2 设 f:A→B,g:B→C 。
（ 1） 若 f,g都是满射,则 gf也是满射；
（ 2） 若 f,g都是单射,则 gf也是单射；
（ 3） 若 f,g都是双射,则 gf也是双射；
F u n c t i o n s
函数证明 （ 2）,对任 a1∈ A,a2∈ A且 a1≠a2,由于 f是单射,故有
f(a1 ) ≠f(a2 )
记 f(a1 ) =b1,f(a2 ) =b2。 又因为 g是单射,所以
g(b1 ) ≠g(b2 )
从而 g f(a1 ) ≠g f(a2 )
F u n c t i o n s
函数定理 3 设 f:A→ B,g:B→ C,h:C→ D
则积运算满足结合律,即
h(gf)=(hg)f
F u n c t i o n s
函数几个常用的函数：
Eigenfunction
floor function
ceiling function
F u n c t i o n s
函数
Eigenfunction:特征函数
F u n c t i o n s
函数
DEFINITION 12,
The floor function assigns to the real number x the
largest integer that is less than or equal to x,The value
of the floor function at x is denoted by x」,The ceiling
function assigns to the real number x the smallest
integer that is greater than or equal to x,The value of
the ceiling function at x is denoted by 「 x
（地板函数） （天花板函数）
F u n c t i o n s
函数关于集合的进一步讨论（基于函数）：
Sequences
string
Language
Countable of elements in sets
F u n c t i o n s
函数
Sequences（序列、顺序）,
A sequence is a function from a subset of the set of
integers ({0,1,2,…} or {1,2,3,…}) to a set V,
an denote the image of the integer n
call an a term of the sequence
a1,a2,a3,……,an
F u n c t i o n s
函数
String（串）
a1a2 a3 …… an n>=0
the length of the string
n=0,empty string
v,symbol list set,alphabet set
F u n c t i o n s
函数
Closure of V（ V的闭包）
V*={a| a=a1a2 a3 …… an 是 V上的串且 n>=0}
Positive Closure of V（ V的正闭包）
V+={a| a=a1a2 a3 …… an 是 V上的串且 n>0}
F u n c t i o n s
函数
Language（语言）
L? V*
F u n c t i o n s
函数从集合的基（ Cardinality)进行 集合的分类
A是有限集,A的基为 n （ n>=0) 或
f:{1,2,…n}?A且 f是一一对应
A是可数无限集,f:N?A且 f是一一对应
A是可数集（ countable),A是有限集或 A是可数无限集
A是不可数集 (uncountable),A不是可数集
F u n c t i o n s
函数小结,
函数的概念函数的分类几个特殊的函数进一步的思考：
一、函数本身作为集合二、集合间的大小比较
F u n c t i o n s
函数练习题：
函数
DEFINITION 1,
1.2.2 函数
Functions
/Mapping 映射
F u n c t i o n s
函数
DEFINITION 1,
Let A and B be sets,A function f from A to B is an
assignment of exactly one element of B to each element
of A,We write f(a) = b if b is the unique element of B
assigned by the function f to the element a of A,If f is a
function from A to B,we write f,A→B,
F u n c t i o n s
函数
DEFINITION 2,
If f is a function from A to B,we say that A is the
domain of f and B is the codomain of f,If f (a) = b,we say
that b is the image of a and a is a pre-image of b,The
range of f is the set of all images of elements of A,Also,if f
is a function from A to B,we say that f maps A to B.
A：定义域 /domain of f a,b 的原象 /pre-image
B：陪域 /codomain of f b,a的象 /image
f（ A）：值域 /range of f
F u n c t i o n s
函数
DEFINITION 3,
Let f1 and f2 be function from A to R,Then f1 + f2
and f1 f2 are also functions from A to R defined by
(f1 + f2 )(x) = f1(x) + f2(x),
(f1 f2)(x) = f1(x) f2(x).
当 B=Real Set
F u n c t i o n s
函数
DEFINITION 4,
Let f be a function from the set A to the set B and let
S be a subset of A,The image of S is the subset of B that
consists of the images of the elements of S,We denote
the image of S by f (S),so that
f (S) = { f (s)∣ s∈ S}.
A的子集 S的象
F u n c t i o n s
函数
DEFINITION 11,
Let f be a function from the set A to the set B,The
graph of the function f is the set of ordered pairs {(a,
b)∣ a∈ A and f (a) = b}.
F u n c t i o n s
函数
EXAMPLE 20
Display the graph of the function f(x) = x2 from the
set of integers to the set of integers.
F u n c t i o n s
函数
DEFINITION 1,
函数的分类
F u n c t i o n s
函数
DEFINITION 5,
A function f is said to be one-to-one,or injective,if
and only f (x) = f (y) implies that x = y for all x and y in
the domain of f,A function is said to be an injection if it
is one-to-one.
一对一，单函数，单射
F u n c t i o n s
函数
EXAMPLE 6
Determine whether the function f
from {a,b,c,d} to (1,2,3,4,5) with
f(a) = 4,f(b) = 5,f(c) = 1,and f(d) = 3
is one-to-one.
F u n c t i o n s
函数
DEFINITION 6,
A function f whose domain and codomain are
subsets of the set of real numbers is called strictly
increasing if f(x) < f(y) whenever x < y and x and y are
in the domain of f,Similarly,f is called strictly
decreasing if f (x) > f (y) whenever x < y and x and y are
in the domain of f.
F u n c t i o n s
函数
DEFINITION 7,
A function f from A to B is called onto,or surjective,
if and only if for every element b∈ B there is an element
a∈ A with f (a) = b,A function f is called a surjection if
it is onto.
映上的，满函数，满射
F u n c t i o n s
函数
EXAMPLE 9
Let f be the function
from {a,b,c,d} to {1,2,3}
defined by f(a) = 3,f(b) = 2,f(c) = 1,and f(d) = 3.
Is f an onto function?
F u n c t i o n s
函数
DEFINITION 8,
The function f is a one-one correspondence,or a
bijection,if it is both one-to-one and onto.
一一对应，双射
F u n c t i o n s
函数
DEFINITION 9,
Let f be a one-to-one correspondence f from the set A
to the set B,The inverse function of f is the function that
assigns to an element b belonging to B the unique
element a in A such that f (a) = b,The inverse function
of f is denoted by f –1,Hence,f –1(b) = a when f(a)=b.
逆函数
F u n c t i o n s
函数定理 1 设 f:A→B 且是 双射,则
f –1,B → A 也是双射
F u n c t i o n s
函数
DEFINITION 10,
Let g be a function from the set A to the set B and let
f be a function from the set B to the set C,The
composition of the functions f and g,denoted by f o g,is
defined by
(f o g)(a) = f (g (a)).
函数的复合运算，积运算
F u n c t i o n s
函数
EXAMPLE 18
Let f and g be the functions from the set of integers to
the set of integers defined by f(x) = 2x + 3 and g(x) = 3x
+ 2,What is the composition of f and g? What is the
composition of g and f?
(f o g )(x) = f(g(x)) = f(3x + 2) = 2(3x + 2) + 3 = 6x + 7
(g o f )(x) = g(f(x)) = g(2x + 3) = 3(2x + 3) + 2 = 6x + 11
F u n c t i o n s
函数定理 2 设 f:A→B,g:B→C 。
（ 1） 若 f,g都是满射,则 gf也是满射；
（ 2） 若 f,g都是单射,则 gf也是单射；
（ 3） 若 f,g都是双射,则 gf也是双射；
F u n c t i o n s
函数证明 （ 2）,对任 a1∈ A,a2∈ A且 a1≠a2,由于 f是单射,故有
f(a1 ) ≠f(a2 )
记 f(a1 ) =b1,f(a2 ) =b2。 又因为 g是单射,所以
g(b1 ) ≠g(b2 )
从而 g f(a1 ) ≠g f(a2 )
F u n c t i o n s
函数定理 3 设 f:A→ B,g:B→ C,h:C→ D
则积运算满足结合律,即
h(gf)=(hg)f
F u n c t i o n s
函数几个常用的函数：
Eigenfunction
floor function
ceiling function
F u n c t i o n s
函数
Eigenfunction:特征函数
F u n c t i o n s
函数
DEFINITION 12,
The floor function assigns to the real number x the
largest integer that is less than or equal to x,The value
of the floor function at x is denoted by x」,The ceiling
function assigns to the real number x the smallest
integer that is greater than or equal to x,The value of
the ceiling function at x is denoted by 「 x
（地板函数） （天花板函数）
F u n c t i o n s
函数关于集合的进一步讨论（基于函数）：
Sequences
string
Language
Countable of elements in sets
F u n c t i o n s
函数
Sequences（序列、顺序）,
A sequence is a function from a subset of the set of
integers ({0,1,2,…} or {1,2,3,…}) to a set V,
an denote the image of the integer n
call an a term of the sequence
a1,a2,a3,……,an
F u n c t i o n s
函数
String（串）
a1a2 a3 …… an n>=0
the length of the string
n=0,empty string
v,symbol list set,alphabet set
F u n c t i o n s
函数
Closure of V（ V的闭包）
V*={a| a=a1a2 a3 …… an 是 V上的串且 n>=0}
Positive Closure of V（ V的正闭包）
V+={a| a=a1a2 a3 …… an 是 V上的串且 n>0}
F u n c t i o n s
函数
Language（语言）
L? V*
F u n c t i o n s
函数从集合的基（ Cardinality)进行 集合的分类
A是有限集,A的基为 n （ n>=0) 或
f:{1,2,…n}?A且 f是一一对应
A是可数无限集,f:N?A且 f是一一对应
A是可数集（ countable),A是有限集或 A是可数无限集
A是不可数集 (uncountable),A不是可数集
F u n c t i o n s
函数小结,
函数的概念函数的分类几个特殊的函数进一步的思考：
一、函数本身作为集合二、集合间的大小比较
F u n c t i o n s
函数练习题：
