Methods of Mathematical Proof

If the proof of a theorem is not immediately apparent, it may be because you are trying the wrong approach. Below are some effective methods of proof that may aim you in the right direction.

1. Proof by Obviousness: "The proof is so clear that it need not be mentioned."
   
2. Proof by General Agreement: "All in Favor?..."
   
3. Proof by Imagination: "Well, We'll pretend its true."
   
4. Proof by Convenience: "It would be very nice if it were true, so ..."
   
5. Proof by Necessity: "It had better be true or the whole structure of mathematics would crumble to the ground."
   
6. Proof by Plausibility: "It sounds good so it must be true."
   
7. Proof by Intimidation: "Don't be stupid, of course it's true."
   
8. Proof by Lack of Sufficient Time: "Because of the time constraint, I'll leave the proof to you."
   
9. Proof by Postponement: "The proof for this is so long and arduous, so it is given in the appendix."
   
10. Proof by Accident: "Hey, what have we here?"
    
11. Proof by Insignificance: "Who really cares anyway?"
    
12. Proof by Mumbo-Jumbo: "For any epsilon> 0 there exists a corresponding delta > 0 s.t. f(x)-L < epsilon whenever x-a < delta."
    
13. Proof by Profanity: (example omitted)
    
14. Proof by Definition: "We'll define it to be true."
    
15. Proof by Tautology: "It's true because it's true."
    
16. Proof by Plagiarism: "As we see on page 238 ..."
    
17. Proof by Lost Reference: "I know I saw this somewhere ..."
    
18. Proof by Calculus: "This proof requires calculus, so we'll skip it."
    
19. Proof by Terror: When intimidation fails ...
    
20. Proof by Lack of Interest: "Does anyone really want to see this?"
    
21. Proof by Illegibility: " ¥ ª Ð Þ þæ"
    
22. Proof by Logic: "If it is on the problem sheet, then it must be true."
    
23. Proof by Majority Rule: Only to be used if General Agreement is impossible.
    
24. Proof by Clever Variable Choice: "Let A be the number such that this proof works."
    
25. Proof by Tessellation: "This proof is just the same as the last."
    
26. Proof by Divine Word: "And the Lord said, 'Let it be true,' and it came to pass."
    
27. Proof by Stubbornness: "I don't care what you say! It is true!"
    
28. Proof by Simplification: "This proof reduces to the statement, 1 + 1 = 2."
    
29. Proof by Hasty Generalization: "Well, it works for 17, so it works for all reals."
    
30. Proof by Deception: "Now everyone turn their backs ..."
    
31. Proof by Supplication: "Oh please, let it be true."
    
32. Proof by Poor Analogy: "Well, it's just like ..."
    
33. Proof by Avoidance: Limit of Proof by Postponement as t approaches infinity.
    
34. Proof by Design: "If it's not true in today's math, invent a new system in which it is."
    
35. Proof by Intuition: "I just have this gut feeling ..."
    
36. Proof by Authority: "Well, Bill Gates says it's true, so it must be."
    
37. Proof by Vigorous Assertion: "And I REALLY MEAN THAT!"
    
38. Proof by A.F.K.T. Theorem: "Any Fool Knows That!"
    
39. Proof by vigorous handwaving: Works well in a classroom.
    
40. Proof by seduction: "Convince yourself that this is true!"
    
41. Proof by accumulated evidence: "Long and diligent search has not revealed a counterexample."
    
42. Proof by Divine Intervention: "Then a miracle occurs ..."

Source: Dick A. Wood in Mathematics Teacher (November 1998).

Stand by Rascal Flatts

You feel like a candle in a hurricane
Just like a picture with a broken frame
Alone and helpless
Like you've lost your fight
But you'll be alright

Cause when push comes to shove
You taste what you're made of
You might bend, till you break
'Cause it's all you can take
On your knees you look up
Decide you've had enough
You get mad you get strong
Wipe your hands shake it off
Then you stand, then you stand

Life's like a novel
With the end ripped out
The edge of a canyon
With only one way down
Take what you're given before it's gone
Start holding on, keep holding on

Everytime you get up
And get back in the race
One more small piece of you
Starts to fall into place

Videotaped Lectures

E. Bennet and N. Maniar mentioned the following potential benefits of videotaped lectures:

"...videoing face-to-face lectures can provide students with a valuable resource to complement their studies. Students can watch the videoed lecture to revisit any points that they did not understand whilst watching the lecture face-to-face. Furthermore, they can stop, start and rewind the video to address their specific needs. In fact, software has been developed that enables students to personalise a videoed lecture by adding their own annotations. On a more practical level, videoing lectures allows students to catch up if they miss a face-to-face lecture. This also enables them to adopt a more flexible learning pattern if they wish...Some (findings) indicate that videoed lectures can improve students’ grades and increase their overall level of satisfaction and confidence with the course."

Read more of it HERE.

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I've posted in a separate blog (Mentor's Notes) some videotaped lectures in mathematics.

The Susan Boyle Phenomenon

Susan Boyle (born 1961) is a Scottish church volunteer and amateur singer who came to public attention when she appeared as a contestant on the third series of Britain's Got Talent. Boyle leapt to almost immediate global fame when she sang "I Dreamed a Dream" from Les Misérables in the competition's first round.

Before she sang, both the audience and the judges appeared to express scepticism based on her age and what was seen as an unattractive appearance. In contrast, her vocal performance was so well received that she has been dubbed "The Woman Who Shut Up Simon Cowell".She received a standing ovation from the live audience, attracting yes-votes from Cowell and Amanda Holden, and the "biggest yes I have ever given anybody" from Piers Morgan. The original talent show and audition was recorded in Scotland in January 2009.

The juxtaposition of the reception to her voice with the audience's first impression of her triggered global interest. Articles about her appeared in newspapers all over the world, while online videos of her performance totalled over 40 million views within a week.Cowell is reported to be setting up a contract with Boyle with his Syco Music company label, a subsidiary of Sony Music.

Boyle is the youngest of nine children and lives in Blackburn with her ten-year-old cat, Pebbles. Boyle has learning disabilities. Her classmates teased her because of this and her appearance.

Early on she received some professional voice training in Livingston, Scotland. She stopped her pursuit of singing to look after her sick mother, who died in 2007 at the age of 91. Her performance in the regional finals of Britain’s Got Talent was the first time Boyle had sung after her mother's death. Boyle stated in The Washington Post that she entered the contest at the behest of her late mother, who urged her to "take the risk" of singing in front of an audience larger than her parish church. She is unmarried and currently unemployed. During her audition video she said she had "never been kissed.",although she later clarified that this had been a joke.She aspires to become a musical theatre singer in the vein of Elaine Paige.

Boyle performed a rendition of "I Dreamed a Dream" from Les Misérables in the first round of the third series of Britain's Got Talent, on 11 April 2009. This performance was widely reported and a video of her singing was viewed by tens of millions of people on the website YouTube. Boyle was reportedly shocked and amazed by the strength of this reaction.

Source: http://en.wikipedia.org/wiki/Susan_Boyle

Different Versions

Tears for Fears




Gary Jules




Adam Lambert




MAD WORLD

All around me are familiar faces
Worn out places, worn out faces
Bright and early for their daily races
Going nowhere, going nowhere
Their tears are filling up their glasses
No expression, no expression
Hide my head I want to drown my sorrow
No tomorrow, no tomorrow

And I find it kinda funny
I find it kinda sad
The dreams in which I'm dying
Are the best I've ever had
I find it hard to tell you
I find it hard to take
When people run in circles
It's a very, very mad world mad world

Children waiting for the day they feel good
Happy Birthday, Happy Birthday
Made to feel the way that every child should
Sit and listen, sit and listen
Went to school and I was very nervous
No one knew me, no one knew me
Hello teacher tell me what's my lesson
Look right through me, look right through me

And I find it kinda funny
I find it kinda sad
The dreams in which I'm dying
Are the best I've ever had
I find it hard to tell you
I find it hard to take
When people run in circles
It's a very, very mad world ... world
Enlarge your world
Mad world

Pics

All God's angels come to us

disguised.

~James Russell Lowell

My sister’s photos taken yesterday during the Easter Vigil (Salubong)

Literature Appreciation 101

There are only two types of readers: those who can "read between the lines" and those who simply "read the lines".

Your reaction to this controversial article by Chip Tsao will be a crude determinant of what type of reader you are.


The War at Home
Chip Tsao

"The Russians sank a Hong Kong freighter last month, killing the seven Chinese seamen on board. We can live with that—Lenin and Stalin were once the ideological mentors of all Chinese people. The Japanese planted a flag on Diàoyú Island. That's no big problem—we Hong Kong Chinese love Japanese cartoons, Hello Kitty, and shopping in Shinjuku, let alone our round-the-clock obsession with karaoke.

"But hold on—even the Filipinos? Manila has just claimed sovereignty over the scattered rocks in the South China Sea called the Spratly Islands, complete with a blatant threat from its congress to send gunboats to the South China Sea to defend the islands from China if necessary. This is beyond reproach. The reason: there are more than 130,000 Filipina maids working as $3,580-a-month cheap labor in Hong Kong. As a nation of servants, you don't flex your muscles at your master, from whom you earn most of your bread and butter.

"As a patriotic Chinese man, the news has made my blood boil. I summoned Louisa, my domestic assistant who holds a degree in international politics from the University of Manila, hung a map on the wall, and gave her a harsh lecture. I sternly warned her that if she wants her wages increased next year, she had better tell every one of her compatriots in Statue Square on Sunday that the entirety of the Spratly Islands belongs to China.

"Grimly, I told her that if war breaks out between the Philippines and China, I would have to end her employment and send her straight home, because I would not risk the crime of treason for sponsoring an enemy of the state by paying her to wash my toilet and clean my windows 16 hours a day. With that money, she would pay taxes to her government, and they would fund a navy to invade our motherland and deeply hurt my feelings.

"Oh yes. The government of the Philippines would certainly be wrong if they think we Chinese are prepared to swallow their insult and sit back and lose a Falkland Islands War in the Far East. They may have Barack Obama and the hawkish American military behind them, but we have a hostage in each of our homes in the Mid-Levels or higher. Some of my friends told me they have already declared a state of emergency at home. Their maids have been made to shout 'China, Madam/Sir' loudly whenever they hear the word 'Spratly.' They say the indoctrination is working as wonderfully as when we used to shout, 'Long live Chairman Mao!' at the sight of a portrait of our Great Leader during the Cultural Revolution. I'm not sure if that's going a bit too far, at least for the time being."

Net Surfing and Productivity

When done for fun, net surfing improves one's level of concentration which leads to increased productivity. That's according to a University of Melbourne study.

Study author Dr. Brent Coker said that workers who engage in "Workplace Internet Leisure Browsing" (WILB) activities such as searching for product info, reading online news, playing online games, watching videos on Youtube, among others, are more productive than those who don't.

“Firms spend millions on software to block their employees from watching videos on YouTube, using social networking sites like Facebook or shopping online under the pretense that it costs millions in lost productivity, however that’s not always the case,” he said.

Dr. Coker, however, added that WILB activities should be done in moderation since internet addiction could have the reverse effect.

Read the full article HERE.

The Most Famous April Fools' Day Hoax of All Time

Happy April Fools' Day everyone!

Based on the info I got from the web, April Fools' Day originated probably in 1582 when Charles IX introduced in France the Gregorian calendar which moved the New Years' Day from April 1 to January 1. Those who did not acknowledge this change were labeled "fools" and practical jokes were played upon them by those who did.

The video clip below shows what's considered to be the "most popular April Fools' Day Hoax of all time". Watch it!