Mathematical Proof Logic

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P. Pooksombat [2011/10/14]

Version 1.0

ppooksombat.blogspot.com

1

1 :

2 : 1

3

, Contradiction Contrapositive

-

- Contradiction

()

- Contrapositive

A B ~B ~A

Without Loss of Generality (WLOG)

(

) WLOG

2

3 : 1

1. 1, 2, 3 3 + 12 + 2 + 3 = 0 1, 2, 3

= max{ 1 , 2 , 3 } < 1 + = 1,2,3

() z 3 = 12 + 2 + 3

||3 = |12 + 2 + 3|

, 12 + 2 + 3 1 2 + 2 + |3|

||3 1 2 + 2 + |3|

= max{ 1 , 2 , 3 } 1 , 2 , 3

||3 2 + +

||3 1 < 3, ||3 1 < { 2 + + 1}

2 + + 1 > 0 1 <

< 1 + #

2. , , , , > 0 + = + = =

(Contradiction) WLOG >

> >

> >

> >

+ > +

() >

>

= #

3

3. 1

A B, B C C A ( 3-cycle)

(Contrapositive) 3-cycle

3-cycle

By Contradiction, 3-cycle

M N k

WLOG M N

M k-1 N k

k N M N M

M k-1 N k

N M M

X X M, M N N X 3-cycle

#

( :

3-cycle

3-cycle

)

4

4. , , 0 + + = 3 + + = 0 4 < 0

, , 3 + + 2 + + + = 0

= = 3 32 , By Contradiction, 0 < 4

0

3 32 0

2( 3) 0

3 , , 3 + + 9 + + = 3

< 4

3 32 < 4

+ 1 2 2 < 0

< 1 , , < 1 + + < 3 + + = 3

4 < 0 #

5. , + + 1 | 2 2 + 2 1 + + 1

( )

By Contradiction, + + 1

2 2 + 2 1 = + 2 + 2 1

= + + 1 ( + 1) + 2

+ + 1 | 2

+ + 1 1

+ + 1 | + + 1 > | |

+ + 1 #

5

6. () [] ( () ) =

=

() []

, < < 0 > 0 () (, )

= 0 (, ) (, ) x

= 0

= 0 < 0 > 0

, WLOG > 0 x

By Contradiction, r = ( )

> 0 >

() () > 0 > ()

= #

( : WLOG > 0 < 0 )

7. , , 3 + 3 + 3 0

3 + 3 + 3 = + + 3 3 + + + + + 3

+ + = 0 3 + 3 + 3 = 3

By Contradiction, 3 + 3 + 3 = 0

+ + = 3 ( )3

0 = 3 ( )3

0 = ( )

= = =

3 + 3 + 3 0 #

6

4 : 2

2

Induction Strong Induction

- (Induction) ()

(1) ()

( + 1) ( (1)

P(2) (3) ... Induction

)

- (Strong Induction) Induction

(1), (2), ()

( + 1) ( (2) (3)

)

() 1

.

7

5 : 2

1. 1 + 2 + 3 + + = +1 2

n

(Induction) () 1 + 2 + 3 + + = +1 2

= 1,2,3,

(1) 1 = 1 2 2

() ( + 1)

1 + 2 + 3 + + = +1 2

+ 1 1 + 2 + 3 + + + + 1 = +1 2

+ + 1

1 + 2 + 3 + + + + 1 = +1 (+2)2

( + 1) () n

1 + 2 + 3 + + = +1 2

n #

(1) ?

(1) (2)

(1) (2) ?

() ( + 1)

(1) (2)

(3) ?

(2)

(3) (3)

(4)

8

2. cos cos n

(Strong Induction) () cos

= 2 = 1

cos 2 = 2 cos2 1 cos

= 3 , cos 3 = 4 cos3 3 cos cos

1 , (2), , () ( + 1)

cos + 1 = cos + = cos k cos sin sin

= cos k cos 12

(cos 1 cos + 1 )

cos( + 1) = 2 cos k cos cos 1

cos , cos 1 , cos cos( + 1)

cos n #

3. ! < 2 !2 2

() ! < 2 !2

(2) 2! < 4!2

2 2 < 6

() ! < 2 !

2

+ 1 ! < ( + 1) 2 !

2

+ 1 = 2 + 2 + 1 < 22 + 3 + 1 = 2+2 (2+1)2

+ 1 ! < 2+2 (2+1) 2 !

22=

2+2 !

2+1

( + 1) ! < 2 !2 2 #

9

6 :

1. 2 0, 1, , > 0 = 1,2, , 1

1 + ( + +1) = 1 +1 2

10

9. 1

12+

1

22+

1

32+ +

1

2 2

1

n

10. 3! | 3 ! 0

11.

12. () [] () = 0,1,2,

( : )

13. + 1

= 2 cos + 1

= 2 cos n

14. 1, 2, , = 1 2 2 2 2 + 1

( :

)

15. = + + 1 + 0 0 0 > 1 + 2 + +

16. P Q P Q

AB A, B P, Q

AB

17. = sin + sin , sine

curve 2 + 2

11

18. (Gamma Function)

= 1

0

1 ! = 1 ! > 0

( : (Integration by Parts), = )

19. cos21 sin21 /2

0=

2 +

(Beta Function)

B , = 1 1 1 1

0

B , = +

, > 0

( : , 0 )

( : = cos2 )

20. = ()

= 2

= = 2

= 2 =

() () = ()

() ( )

( )

= (()) = (()) ()

() = ()

21. 1 = 2/2 2 =

2/2 ,

= 12

(x2

2

2 )

12

22. sin + sin 2 + + sin =sin

2sin +1

2

sin2

23. cos + cos 2 + + cos =sin

2cos +1

2

sin2

24. (Bonus) 23 sin2 + sin2 2 + + sin2 cos2 +cos2 2 + + cos2

25. (Bonus) 1 1 1 ( 51

2 )

-

2011-10-15T02:04:09+0700Perathorn Pooksombat