matlab 座標軸設定

axis( [ 0.0 2.0 0.0 2.5 ]);

座標軸x在0.0和2.0之間
座標軸y在0.0和2.5之間

記得不要加逗號

Matlab不定積分

syms y x a b

y=a*x^2+b*x;

int(sqrt(1+diff(y)^2));


log(b + ((b + 2*a*x)^2 + 1)^(1/2) + 2*a*x)/(4*a) + (((b + 2*a*x)^2 + 1)^(1/2)*(b + 2*a*x))/(4*a)


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syms y x a b 變數

int 積分


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要用到matlab symbolic math toolbox

如何在多個點畫出凸多邊形

如何在多個點畫出凸多邊形

convex hull演算法

http://www.csie.ntnu.edu.tw/~u91029/ConvexHull.html

這個網頁說明很詳細

離散數學(2)-1

Show that for primitive statements p,q

p V q(p xor q) ↔ (p^~q)V(~p^q) ↔ ~(p ↔ q)

p	q	p V q	p^~q	~p^q	(p^~q)V(~p^q)	p → q	q → p	p↔q	~ p↔ q
0	0	0	0	0	0	1	1	1	0
0	1	1	0	1	1	1	0	0	1
1	0	1	1	0	1	0	1	0	1
1	1	0	0	0	0	1	1	1	0

離散數學(8)

第八章 The Principle of Inclusion and Exclusion

The Principle of Inclusion and Exclusion. Cosider a set S, with |S|=N, and coditions ci, 1<=i<=t, each of which way be satisfied by some of the elements of s of S. The number of elements of S that satisfy none of the conditions ci, 1<=i<=t, is denoted by N'=N(c1'c2'c3'...ct') where

N'=N-[N(c1)+N(c2)+...+N(ct)]+[N(c1c2)+n(c1c3)+...+N(c1ct)+N(c2c3)+...+N(ct-1ct)]-[N(c1c2c3)+N(c1c2c4)+...+N(c1c2ct)_N(c1c3c4)+...+N(c1c3ct)+...+N(ct-2ct-1ct)]+...+
(-1)^tN(c1c2c3...ct)

N'=N-1<=i<=t∑N(ci)+1<=i<j<=t∑N(cicj)-1<=i<j<k<=t∑N(cicjck)+...+
     (-1)^tN(c1c2c3...ct)

離散數學(7)

第七章 Relations: The Second Tine Around

A relation R on a set A is called reflexive(反身性) if for all x ε A (x,x) ε R

Relation R on set A is called symmetric(對稱性) if(x,y) εR → (y,x) ε R for
all x,y ε A

For a set A, a relation R on A is called transitive of, for all x,y,z ε A, (x,y),(y,z) ε R → (x,z) ε R.(So if x “is related to” y, and y “is related to” z, we want x “related to” z, withy playing the role of “intermediary”.

Given a relation R on a set A, R is called antisymmetric if for all a,b εA ,(aRb and bRa) → a=b

A relation R on a set A is called a partial order, or a partial ordering relation, if R is reflexive, antisymmetric, and transitive.

If A, B, and C are sets with R1cAxB and R2c BxC, then the composite relation R1◦R2 is a relation from A to C defined by  R1◦R2={(x,z)|x εA,
z ε C, and thereexists y ε B with (x,y) εR (y,z) εR2}

長方桌排列

有10個人 在長方桌作排列

一邊可坐3人 一邊可坐2人

共有10! /2種排列

離散數學(6)

第六章 Languages: Finite State Machines

If ∑ is an alphabet and n ε Z+, we define the powers of ∑ recursively as follows:

1)∑1=∑

2)∑^n+1={xy|x ε ∑, y ε ∑^n} where xy denote the juxtaposition of x and y

For a alphabet ∑ we define ∑0={λ},where λ denotes the empty string—that is, the string consisting of no symbols taken from ∑.

If ∑ is an alphabet

a)∑ +=n=1 to 無限大U∑ n=Un ε z+∑ n

b)∑ *=n=0 to無限大 U∑ n

離散數學(5)

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.

離散數學(4)

第四章 Properties of the integers: Mathematical Induction

All that is needed is for the open statement S(n) to be true for some first element n0 ε Z so that the induction process has a starting place. We need the truth of S(n0) for our basis step. The integer n0 could be 5 just as well as 1. It could even be zero or negative because the set Z+ in union with {0}{ or any finite set of negative integers is well ordered.

Principle of Mathematical Induction

[S(n0) ^ [all k>=n0[s(k) → s(k+1)]]] → all n >=no s(n)

Principle of Strong Mathematical Induction

a) If s(n0),s(n0+1), s(n0+2)...,s(n1-1),s(n1) are true

b)If whenever s(n0) s(n0+1)...s(k-1)and s(k) are true for some k ε z+, where k>=n1, then the statement s(k+1) is also true;then s(n) is true for all n>=n0

If a,b ε z and b=\0 , we say that b divides a and we write b|a, if there is an integer n such that a=bn, where this occurs we say that b is a divisor(因數), or a is a multiple(倍數) of b.

a) 1|a and a|0

b) [(a|b)^(b|a)] → a=+-b

c)[(a|b)^(b|c)] → a|c

d) a|b → a|bx for all x ε Z

e) If x=y+z for some x,y,z ε Z, and a divides two of the three integers x,y and z, then a divides the remaining integer.

f)[(a|b)^a|c)] → a|(bx+cy) for all x,y ε Z

g)For 1<=i<=n let ci ε Z. If a divides each ci, then a|(c1x1+c2x2+...+cnxn), where xi ε Z for all 1<=i<=n

離散數學(3)

第三章 Set Theory

A set should be well-defined collection of objects. These objects are called elements and we said to be members of the set.

If C,D are sets from a universe U, we say that C is a subset of D and write C c D or D c C , if every element of C is an element of D. If , in addition, D contains an element that is not in C, then C is called a proper subset of D, and this is denoted by C c D

for A,B c U we define the flowing

a) A U B( the union of A and B) = {x|x ε A V ε B}

b) A∩B( the intersection of A and B) = {x| x ε A ^ ε B}

c)AΔB(the symmetric difference of A and B ) = {x| (x ε A V ε B) ^

x not ε A∩B}={x|x ε AUB ^ x not ε A∩B}

離散數學(2)

第二章 Fundamentals of Logic

In the development of any mathematical theory, assertions are made in the form of sentences. Such verbal or written assertions, called statements (or propositions), are declarative sentences that are either true or false.

1)Transform a given statement p into the statement ¬p, which denotes its negation and is read “Not p.”

2)

a) Conjunction:The conjunction of the statements p,q is denoted by p^q, which is read “p and q.”

b) Disjunction: The expression p V q denotes the disjunction of the statements p,q and is read “p or q.”

c)Implication: We say that “p implies q” and write p->q to designate the statement, which is the implication of q by p.

i)If p, then q.

ii)p is sufficient for q

iii)p is sufficient condition for q

iv)q is necessary for p
v)q is a necessary condition for p
vi) p only if q
d) Bi-conditional: The bi-conditional of two state¬ments p,q is denoted by
p ↔ q, which is read “p if and only if q” or “p is necessary and sufficient for q”
“p if and only if q” 可縮寫成 “p iff q”

p q p^q p V q p V q p->q q ↔ p

0 0 0 0 0 1 1

0 1 0 1 1 1 0

1 0 0 1 1 0 0

1 1 1 1 0 1¬ 1

A compound statement is called a tautology if it is true for all truth value assignments for its component statements. If a compound statement is false for all such assignments, then it is called a contradiction.

¬

Two statements s1,s2 are said to be logically equivalent, and we write

s1 ↔ s2, when the statement s1 is true if and only if the statement s2 is true.

¬

Let s be a statement. If s contains no logical connectives other than ^ and V, then the dual of s denoted sd, is the statement obtained from s by replacing each occurrence of ^ and V by V and ^, respectively, and each occurrence of T0 and F0 by F0 and To, respectively.

The principle of duality, let s and t be statements that contain no logical connectives other than ^ and V.

if s ↔ t, than sd ↔ td

p->q ↔ ~p V q

The statement ¬q → ¬p is called contrapositive of the implication p → q

The statement q → p is called the converse of p → q

¬p → ¬q is called the inverse of p → q

The contrapositive(否逆)：¬q → ¬p

The converse(逆): q → p

The inverse (否) : ¬p → ¬q

When we write allxp(x)->exist x p(x) we are saying that the implication allxp(x) → exist x p(x) is a logical implication.


Two substitution rules
1)Suppose that the compound statement P is a tautology. If p is primitive statement that appears in P and we replace each occurrence of p by the same q, then the resulting compound statement P1 is also a taotology.
2)Let P be a compound statement where p is an arbitrary statement that appears in P, and let q be a statement such that q ↔ p, Suppose that in P we replace one or more occurences of p by q. Then this replacement yields the compound statement P1. Under these circumstances P1 ↔ P.

p → q   <=> ~p V q  <=> ~q → ~p

離散數學(1)

第一章 Fundamental Principles of Counting

Enumeration, or counting, may strike one as an obvious process that a student learns when first studying arithmetic.

Our study of discrete and combinatorial mathematics begins  with two basic principles of counting: the rules of sum and product.

The Rule of Sum: If a first task con be performed in m ways, while a second task can be performed in n ways, and the two task cannot be performed simultaneously, then performing either task can be accomplished in any one of m+n ways.

The Rule of Product: If a procedure can be broken down into first and second stages, and if there are m possible outcomes for first stage and if, for each of these outcomes, there are n possible outcomes for the second stage, then the total procedure can be carried out, in the designated order, in mn ways.

For an integer n >= 0, n factorial ( denoted n¦)
is defined by 0¦=1
                         n¦=(n)(n-1)(n-2)...(3)(2)(1) for n >= 1

Given a collection of n distinct objects, any(linear) arrangement of these objects is called a permutation of the collection.

If there are n distinct objects and r is an integer, with 1<=r<=n, then by the rule of product, the number of permutation of size r for the n objects

p(n,r)=n*(n-1)*(n-2)*...*(n-r+1)
=(n)(n-1)(n-2)...(n-r+1)*(n-r)(n-r-1)...(3)(2)(1)/((n-r+1)*(n-r)(n-r-1)...(3)(2)(1))
=n¦/(n-r)¦

If there are n objects with n1 indistinguishable objects of a first type, n2  indistinguishable objects of a second type,..., and nr  indistinguishable objects of a nth type, where n1+n2+...+nr=n
then there are n¦/(n1¦n2¦...nr¦) (linear) arrangements of the given n objects.

If we start with n distinct objects, each selection, or combination, of r of these objects, with no reference to order,corresponds to r¦ permutations of size r from the n objects. Thus the number of combinations of size r from a collection of size n is

C(n,r)=P(n,r)/r¦=n¦/(r¦(n-r)¦)  0<=r<=n

The binomial Theorem
(x+y)^n=(n,0)x^0y^n+(n,1)X^1y^n-1+(n,2)X^2y^(n-2)+...+(n,n-1)x^(n-1)y^1+(n,n)x^ny^0=form k=0 to n∑(n,k)x^ky^(n-k)

線性代數

線性代數為linear system
但是現實生活為non linear system
也就是現實生活有一個數學模型來解決這個現實生活的問題