§4 谓词演算的性质 - 复旦大学精品课程.
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Let f={(-2,0),(-1,1), (0,0),(1,3),(2,5)}. f is a function. Rf=? X={-2,0,1}, f(X)=? Y={0,5}, f -1(Y)=?
Theorem 3.1: Let f be a function from A to B, and A1 and A2 be subsets of A. Then
(2)For every aA, If there exist x,yC such that (a,x) f gand (a,y) f g，then x=y?
Definition 3.4: Let g be a function from A to B, and f be a function from B to C. Then composite relation f g is called a function from A to C, we write f g:A→C. If aA, then(f g)(a)=f(g(a)).
and f be a function from B to C. Then composite relation f g is a function from A to C.
Proof: (1)For any aA, there exists cC such that (a,c) f g?
Example：A1={1,2,3,4,5,6},(A1,) 1 is a minimal element of A1 6 is a maximal element of A1
(A1,|) 1 is a minimal element of A1. As these example shows, a poset
The definition of one to one may be restated in the following equivalent form:
If f(a1)=f(a2) then a1=a2 for all a1, a2A Or If a1a2 then f(a1)f(a2) for all a1, a2A
b=b'
Example：Let A={1,2,3,4},B={a,b,c},
R1={(1,a),(2,b),(3,c)}, R2={(1,a),(1,b),(2,b),(3,c),(4,c)}, R3={(1,a),(2,b),(3,b),(4,a)} Example: Let A ={-2,-1, 0,1,2} and B={0,1,2,3,4,5}.
Note: difference between greatest element and maximal element
Example：A1={1,2,3,4,5,6},(A1,) 1 is the least element of A1. 6 is the greatest element of A1. (A1, |) :1 is the least of A1. There is no greatest element.
(1)If A1A2, then f(A1) f(A2)
(2) f(A1∩A2) f(A1)∩f(A2) (3) f(A1∪A2)= f(A1)∪f(A2)
(4) f(A1)- f(A2) f(A1-A2)
Proof: (3)(a) f(A1)∪f (A2) f(A1∪A2) (b) f(A1∪A2) f(A1)∪f (A2)
Definition3.1: Let A and B be nonempty sets. A relation is a function from A to B, denoted by f : AB, if for every aA, there is one and only b B so that (a,b) f, we say that b=f (a). The set A is called the domain of the function f. If XA, then f(X)={f(a)|aX} is called the image of X. The image of A itself is called the range of f, we write Rf. If YB, then f -1(Y)={a|f(a)Y} is called the preimage of Y. A function f : AB is called a mapping. If (a,b) f so that b= f (a), then we say that the element a is mapped to the element b.
3)Let h:Z→Zm={0,1,…m-1}, h(a)=a mod m
onto ,one to one?
3.2 Composite functions and Inverse functions posite functions Relation ,Composition, Theorem3.3: Let g be a function from A to B,
(4) f(A1)- f(A2) f(A1-A2) for any y f(A1)-f(A2)
Theorem 3.2：Let f be a function from A to B, and AiA(i=1,2,…n). Then
n
n
Байду номын сангаас
(1) f ( Ai ) f ( Ai )
Theorem 3.5: Let f be a function from A to B. Then
(i)fIA=f. (ii) IBf = f. Proof. Concerning(i), let aA, (fIA)(a) ?=f(a). Property (ii) is proved similarly to property (i).
Since composition of relations has been shown to be associative (Theorem 2.), we have as a special case the following theorem.
Theorem 3.4: Let f be a function from A to B, and g be a function from B to C, and h be a function from C to D. Then h(gf )=(hg)f
can have more than one maximal element and more than one minimal element.
Definition 2.25: Let (A, ≼) is a poset. An elements aA is called a greatest element of A if x≼a for all xA. An elements aA is called a least element of A if a≼x for all xA.
i 1
i 1
n
n
(2) f ( Ai ) f ( Ai )
i 1
i 1
2. Special Types of functions Definition 3.2：Let A be an arbitrary nonempty set.
The identity function on A, denoted by IA, is defined by IA(a)=a. Definition 3.3.: Let f be a function from A to B. Then we say that f is onto(surjective) if Rf=B. We say that f is one to one(injective) if we cannot have f(a1)=f(a2) for two distinct elements a1 and a2 of A. Finally, we say that f is one-to-one correspondence(bijection), if f is onto and one-toone.
A2={2,3,6,12,24,36},(A2,|) There is no greatest element. There
is no least element.
Definition 2.26: Let (A, ≼) is a poset, and BA. An element aA is called an upper bound of B if b≼a for all bB. An element aA is called a lower bound of B if a≼b for all bB.
RA×B,R is a relation from A to B， DomRA。
(a,b)R (a, c)R
DomR=A, (a,b)R (a, c)R unless b=c function。
Chapter 3 Functions
3.1 Introduction
(1)Domf=A； (2)if (a,b) and (a,b')f, then b=b‘ Relation: DomRA function: DomR=A Relation: (a,b),(a,b')R, but function : if (a,b) and (a,b')f, then
3.Extremal elements of partially ordered sets
Definition 2.24: Let (A, ≼) is a poset. An elements aA is called a maximal element of A if there is no elements c in A such that a≺c. An elements bA is called a minimal element of A if there is no elements c in A such that c≺b.
Example:1) Let f:R(the set of real numbers)→C(the set of complex number), f(a)=i|a|;
2)Let g: R(the set of real numbers)→C(the set of complex number), g(a)=ia;
Example: A2={2,3,6,12,24,36},(A2,|) P={2,3,6},
all upper bounds of P are
P has no lower bounds.
Definition 2.27: Let (A,≼) is a poset, and BA. An element aA is called a least upper bound of B, (LUB(B)), if a is an upper bound of B and a≼a’, whenever a’ is an upper bound of B. An element aA is called a greastest lower bound of B, (GLB(B)), if a is a lower bound of B and a’≼a, whenever a’ is an lower bound of B.
Theorem 3.1: Let f be a function from A to B, and A1 and A2 be subsets of A. Then
(2)For every aA, If there exist x,yC such that (a,x) f gand (a,y) f g，then x=y?
Definition 3.4: Let g be a function from A to B, and f be a function from B to C. Then composite relation f g is called a function from A to C, we write f g:A→C. If aA, then(f g)(a)=f(g(a)).
and f be a function from B to C. Then composite relation f g is a function from A to C.
Proof: (1)For any aA, there exists cC such that (a,c) f g?
Example：A1={1,2,3,4,5,6},(A1,) 1 is a minimal element of A1 6 is a maximal element of A1
(A1,|) 1 is a minimal element of A1. As these example shows, a poset
The definition of one to one may be restated in the following equivalent form:
If f(a1)=f(a2) then a1=a2 for all a1, a2A Or If a1a2 then f(a1)f(a2) for all a1, a2A
b=b'
Example：Let A={1,2,3,4},B={a,b,c},
R1={(1,a),(2,b),(3,c)}, R2={(1,a),(1,b),(2,b),(3,c),(4,c)}, R3={(1,a),(2,b),(3,b),(4,a)} Example: Let A ={-2,-1, 0,1,2} and B={0,1,2,3,4,5}.
Note: difference between greatest element and maximal element
Example：A1={1,2,3,4,5,6},(A1,) 1 is the least element of A1. 6 is the greatest element of A1. (A1, |) :1 is the least of A1. There is no greatest element.
(1)If A1A2, then f(A1) f(A2)
(2) f(A1∩A2) f(A1)∩f(A2) (3) f(A1∪A2)= f(A1)∪f(A2)
(4) f(A1)- f(A2) f(A1-A2)
Proof: (3)(a) f(A1)∪f (A2) f(A1∪A2) (b) f(A1∪A2) f(A1)∪f (A2)
Definition3.1: Let A and B be nonempty sets. A relation is a function from A to B, denoted by f : AB, if for every aA, there is one and only b B so that (a,b) f, we say that b=f (a). The set A is called the domain of the function f. If XA, then f(X)={f(a)|aX} is called the image of X. The image of A itself is called the range of f, we write Rf. If YB, then f -1(Y)={a|f(a)Y} is called the preimage of Y. A function f : AB is called a mapping. If (a,b) f so that b= f (a), then we say that the element a is mapped to the element b.
3)Let h:Z→Zm={0,1,…m-1}, h(a)=a mod m
onto ,one to one?
3.2 Composite functions and Inverse functions posite functions Relation ,Composition, Theorem3.3: Let g be a function from A to B,
(4) f(A1)- f(A2) f(A1-A2) for any y f(A1)-f(A2)
Theorem 3.2：Let f be a function from A to B, and AiA(i=1,2,…n). Then
n
n
Байду номын сангаас
(1) f ( Ai ) f ( Ai )
Theorem 3.5: Let f be a function from A to B. Then
(i)fIA=f. (ii) IBf = f. Proof. Concerning(i), let aA, (fIA)(a) ?=f(a). Property (ii) is proved similarly to property (i).
Since composition of relations has been shown to be associative (Theorem 2.), we have as a special case the following theorem.
Theorem 3.4: Let f be a function from A to B, and g be a function from B to C, and h be a function from C to D. Then h(gf )=(hg)f
can have more than one maximal element and more than one minimal element.
Definition 2.25: Let (A, ≼) is a poset. An elements aA is called a greatest element of A if x≼a for all xA. An elements aA is called a least element of A if a≼x for all xA.
i 1
i 1
n
n
(2) f ( Ai ) f ( Ai )
i 1
i 1
2. Special Types of functions Definition 3.2：Let A be an arbitrary nonempty set.
The identity function on A, denoted by IA, is defined by IA(a)=a. Definition 3.3.: Let f be a function from A to B. Then we say that f is onto(surjective) if Rf=B. We say that f is one to one(injective) if we cannot have f(a1)=f(a2) for two distinct elements a1 and a2 of A. Finally, we say that f is one-to-one correspondence(bijection), if f is onto and one-toone.
A2={2,3,6,12,24,36},(A2,|) There is no greatest element. There
is no least element.
Definition 2.26: Let (A, ≼) is a poset, and BA. An element aA is called an upper bound of B if b≼a for all bB. An element aA is called a lower bound of B if a≼b for all bB.
RA×B,R is a relation from A to B， DomRA。
(a,b)R (a, c)R
DomR=A, (a,b)R (a, c)R unless b=c function。
Chapter 3 Functions
3.1 Introduction
(1)Domf=A； (2)if (a,b) and (a,b')f, then b=b‘ Relation: DomRA function: DomR=A Relation: (a,b),(a,b')R, but function : if (a,b) and (a,b')f, then
3.Extremal elements of partially ordered sets
Definition 2.24: Let (A, ≼) is a poset. An elements aA is called a maximal element of A if there is no elements c in A such that a≺c. An elements bA is called a minimal element of A if there is no elements c in A such that c≺b.
Example:1) Let f:R(the set of real numbers)→C(the set of complex number), f(a)=i|a|;
2)Let g: R(the set of real numbers)→C(the set of complex number), g(a)=ia;
Example: A2={2,3,6,12,24,36},(A2,|) P={2,3,6},
all upper bounds of P are
P has no lower bounds.
Definition 2.27: Let (A,≼) is a poset, and BA. An element aA is called a least upper bound of B, (LUB(B)), if a is an upper bound of B and a≼a’, whenever a’ is an upper bound of B. An element aA is called a greastest lower bound of B, (GLB(B)), if a is a lower bound of B and a’≼a, whenever a’ is an lower bound of B.