(Math Foundation 笔记)1.实数公理和用公理证明3

Using transitivity in a proof：传递性证明的小技巧

If we need to demonstrate an inequality, like A<B

1.      Show A<C for some C

2.      Show C <B

3.      By transitivity, therefore A<B

Example:

·        For all a b x y in R set if a<b and x<y than a+x < b+y

Proof:

Assume a<b and x<y

a+x<b+x by O3&& based on a<b

x+b<y+b byO3 && based on x<y

a+x<b+Y byO2(transitivity)

therefore…

·        For all x y in R set, if 0<x and x<y ,thenx^2 < y^2

Assume 0<x and x<y

x*x<y*x by O4 based on x<yand x>0

x^2<yx

y>0 byO2 since 0<x andx<y

x*y <y*y by O4 and based ony>0

yx<y^2

x^2<y^2 by O2(transitivity)

therefore…

 

another different type ofexample:

·        For all x in R set if x^2<=x then x<=1(suppose use the equivalent  Boolean exp:if p then z == NEG(p) OR z)

 

Using the concept of contrapositive:

If p then z is equivalent to if NEG(Z) thenNEG(p)

 

So we can change statement to For all x inR set if x>1 then x^2>x

Assume x>1

Since 1>0 by pervious prop.

x>0 by O2(tran)

x*x>1*x by O4

x^2>x

therefore…

 

 

另一部分：

direct proof review：

 

to prove all x in set U, p(x)

let x in set U

//x is arbitrary value

Demonstrate p(x)

 

To prove at least x in set U, p(x)

Put x to assign a specific value

Demonstrate p(x) //where x is assigned

 

直接证明：当if语句有全称量词时...

Example:

 All ain set R if all x in set R ax=a then a=0

 

Proof let all a in set R

Assume all x in set R,ax=a

Since 0 is in set R a*(0)=a

Therefore,0=a

Threrfore…

 

***在if statement 里如果有全称量词，因为证明时if条件是假设为True的 所以可以举例一个x使其满足--（这个不好理解啊--）

 

                All a in set R if all x in set R,ax<=x, then a=1

                Proof leta in set R

                Assume all x in set R ax<=x

                Since a*1<=1 ;have a<=1

                Since a*(-1)<=-1;-a+(a+1)<=-1+a+1

Have 1<=a

Now since a<=1 and 1<=a by trichotomy  this imples a=1

Therefore…

All a in set R if at least x in set R,x!=0 and ax=x thena=1

Proof: let a in set R

Assume at least x in set R x!=0 and ax=x

Choose b in set R with b!=0 and ab=b

Then abb^-1=b^-1 so a=1

Therefore…

 

All a in set R if at least x in set R a+x=x then all y inset R ay=a

Proof let all a in set R

Assume at least x in set R, a+x=x

Choose b in set R with a+b=b

Let all y in set R

Since a+b=b, (a+b)-b=b-b

So, a=0

Therefore ay=0*y=0=a

Therefore…

 

All a in set R if all x in set R,a+x=x then at least y inset R,ay=a

Proof let all a in set R

Assume all x in set R,a+x=x

//since 0 inset R a+0=0

//so, a=0

Put y=1

Then ay=a(1)=a

Therefore…

 

总结：

·        if->全程量词 ，假设时举例套用

·        If->存在量词，假设时选一个代表变量

·        Then->全局量词，相当于选代表 但是用原变量就行

·        Then->存在变量，选一个合适的特定的数值赋值给该变量