Proof Set
Proof Set
Proof by Contradiction
The phrase "reductio ad absurdum" literally means reduction to absurdity, and it describes an ancient method of proof which we tend now to call by the name of "Proof By Contradiction."
In this method of proof, we take the proposition we are trying to prove, and assume that the opposite is true. Then we follow a logical argument until we reach a conclusion which contradicts our original assumption. Then we know that the assumption was false, and so we have proved what we wanted to prove.
There are many very beautiful and elegant proofs which use this method. One of my favorites is the Proof of the Infinitude of Primes, given by Euclid. I shall try to present the proof here without using any too technical language!
Before we start, we have to talk about numbers. Everyone knows that if you start counting from the number 1 you could go on for ever without stopping. The way we describe this is that the set of counting numbers is an infinite set. There is no number of which you can say, this is the biggest one, we've now reached the end!
There are some special numbers in the set that have a peculiar property. These are called the Prime Numbers. A prime number can only be divided by itself or the number 1.
For example, 17 is only divisible by 17 or 1, whereas a number like 15 is divisible by 1, 3, 5, and 15 itself. So 17 is a prime, but 15 is not.
Now the question arises, is the set of prime numbers an infinite set? Do the prime numbers go on for ever, or is there a number to be found of which we can say, this is the largest prime number, and there are no more after this?
At first sight, you might think that there IS a largest prime, because as you count higher and higher, there are more smaller numbers that can potentially divide into the number you have reached. However, Euclid believed that they went on for ever, that there was an infinitude of primes. And he proved it, using reductio ad absurdum, proof by contradiction.
First, assume that there is a largest prime number, we give it the label p, and this number is divisible only by itself and 1.
Now we construct a (much larger) number by multiplying together all the prime numbers from 2 up to our number p. We label this number N.
Now, by definition, N is divisible by 1, 2, 3, and all the prime numbers up to and including the number p.
Now consider the number that follows the number N, we will label it (N+1). This number is certainly divisible by 1 and itself, as all numbers have that property. But it is not divisible by 2, since the previous number N is divisible by 2. Also, for the same reason, (N+1) is not divisible by 3, 4, 5, and so on all the way up to N, the number just before it.
Therefore we are forced to conclude that (N+1) is a prime number. But this contradicts our original assumption, since (N+1) is larger than p, and you remember that we assumed p to be the largest prime. Therefore our assumption must be false and hence there is no largest prime number.
Thus we have proved that the set of prime numbers is an infinite set.
Author's Note: The article above was first published by me on Qassia. To find out more about Qassia, and what makes it different from other sites, please visit my Qassia page on http://zencath.qassia.com/
About the Author
Robert Paterson writes on a wide variety of subjects. Currently he is involved with a project designed to raise awareness of the benefits of water filters in the home. You are invited to visit his blog http://purewaterstore.blogspot.com/ to find out more about this.
Proof that Most Haunted is Set Up
 |
 Singer Futura Sewing Embroidery Machine CE150  US $519.00
|
 2 PERSON SAUNA MADE IN USA Carbon FAR Infrared Hemlock Lifetime Warranty New US $1,694.87
|
 Jiffy Proline Commercial Garment Steamer PINK J4000 US $249.00
|
 3 PERSON SAUNA MADE IN USA Carbon FAR Infrared Cedar Lifetime Warranty New US $2,058.87
|
 2000 S 90 SILVER MASSACHUSETTS GEM PROOF No Rerseve US $7.49
|
 Harry Potter and the Deathly Hallows by J K Rowling 2007 Hardcover US $15.00
|
 Casio GzOne Boulder Mint No Contract Ruggedize VERIZON US $64.95
|
 Singer Curvy Sewing Machine 8763 US $199.00
|
 Montel Williams Living Well Set of 3 15oz Travel Tumblers US $19.99
|
 2 x Ultra Clear LCD Screen Protector Guard Film for Amazon Kindle Fire US $3.29
|
 Singer Quantum Sewing Machine 7470 US $369.00
|
 Singer 7469Q Confidence Quilter Model Sewing and Quilting Machine US $299.99
|
 VINTAGE Gres Cabochard Perfume Purse Spray w 3 RefillsHard2FindNIB US $187.50
|
 Singer Curvy Sewing Machine 8770 US $499.00
|
 HAGRID AND Norberta AND CROSSBOW FROM HARRY POTTER HAGRIDS HUT LEGO 4738 US $9.99
|
 Carlos Pena 2008 Topps Stadium Club Photographers Proof 49 50 Tampa Bay Rays US $.99
|
 ARAGOG SPIDER AND 3 BABY SPIDERS FROM HARRY POTTER HAGRIDS HUT LEGO SET 4738 US $9.99
|
 Set of 3x Duck Figurines Ornament with Gloss Finish US $8.08
|
 Brondell Swash 900 Bidet Toilet Seat ROUND or ELONGATED WHITE S900 RW or EW US $335.00
|
 2007 S GEM PROOF DCAM SACAGAWEA DOLLAR FROM US PROOF SET 1 COIN US $2.31
|
 Lot of 2 Mally Beauty Evercolor Automatic Waterproof Eyeliner 24K Tinsel US $14.99
|
 LUCIUS MALFOY DEATH EATER MAGIC WAND LEGO fiG from 4867 HOGWARTS HARRY POTTER US $9.99
|
 VINTAGE Molinard de Molinard Eau De Toilette Perfume Spray1 OZUniqueNIB US $17.49
|
 1999 SILVER PROOF SETS GEM FROSTED CAMEO PROOF IN ORIGINAL US MINT PACKAGING US $129.99
|
 1964 Philadelphia proof set in plastic US $19.48
|
 2006 SCORE SELECT MIKE DOSS ARTISTS PROOF ED 01 32 US $.99
|
 FRENCH POLYNESIA Ile Crescent Crescent Island 1000 Poa Fantasy Coin US $18.99
|
 2000 S 90 SILVER SOUTH CAROLINA GEM PROOF No Rerseve US $7.49
|
 1961 United States Mint Silver Proof Set US $24.00
|
 Honda Odyssey LX 2000 HONDA ODYSSEY LX 1 OWNER GEORGIA OWNED RUST FREE ABSOLUTELY NO RESERVE US $3,550.00
|
 AWESOME DOUBLE SIDE RAINBOW JEFFERSON NICKEL 1938 P NGC MS66 US $23.15
|
 2008 US Mint Proof Set 14 coins w box COA includes 10th year state quarters US $42.00
|
 NobleSpirit GEM PROOF 1986 PROOF SET w Orig Package US $5.05
|
 KARAOKE SYSTEM CAVS PLAYER 203G USB SPEAKERS CDG MUSIC AMERERICAN DJ PPA 210 US $1,299.00
|
 RARE 1991 NOVA EUROPAE International Numismatic Congress Proof Coin Set US $75.00
|
 85 93 BMW E30 3 SERIES 2 4DR FRONT BUMPER PROJECTOR FOG LIGHTS DRIVING LAMPS SET US $71.49
|
 2011 American Silver Eagle 25th Anniversary Set MS PF 69 Mercanti Label FS US $597.22
|
 1935 P Good Buffalo Indian Head Nickel Coin US Mint Coins Coinhut4599 US $.99
|
 Australia Decimal Set of 2 Youth Arts Council Dance Music 1998 Used US $.69
|
 Three United States Uncirculated Mint Sets 1968 including Kennedy Half Dollar US $19.00
|
| Powered by phpBay Pro |
|
|
Thermos Foogo Leak-Proof Stainless Steel 10-Ounce Food Jar, Pink $9.75 Foogo Vacuum Insulated Food JarTherMax double wall vacuum insulation for maximum temperature retention Unbreakable stainless steel interior and exterior Wide mouth is easy to fill, eat from, and clean Ergonomically designed lid with rubber grip Dishwasher safe Non-slip, scratch resistant base, cool to the touch with hot foods Light, compact and portable... |
|
|
Thermos Stainless King SK3000MBTRI4 Food Jar, Midnight Blue $18.99 Thermos SK3000MB4 Food Jar SK3000MB4 Thermos... |
|
|
EasyLunchboxes 3-compartment Bento Lunch Box Containers (Set of 4) BPA-Free. Easy-Open Lids (Not Leakproof) $13.95 Save time, money, paper, and plastic when you use EasyLunchboxes. No more wasteful baggies or mismatched plastic containers. Now you can pack lunches fast for work or school. Even make homemade "Lunchables®!" Created by a mom of three, these bento-style, 3-compartment containers are designed to be carried upright in the EasyLunchboxes cooler bag (choose from 7 vibrant colors). Container lids stay... |
|
|
Priva Ultra Plus Waterproof Sheet Protector, 34 x 52, White $14.99 Keeps sheets dry all night. Unique brushed polyester fabric channels fluid away from the surface for maximum comfort and ultra-dry surface. Absorbs quickly and locks in moisture for the best in skin care. Goes over the bottom sheet to protect mattress AND sheets from moisture and staining. No more middle-of-the-night bed stripping, simply replace the Priva sheet protector with a fresh one in secon... |
|
|
Pampers Splashers ... |
|
|
Tweezerman His Stainless Nail Clipper Set $7.25 Extra durable, heavy duty clippers specially designed for men. Fingernail and toenail clippers are packed together for added convenience. Both clippers feature sharp, strong cutting edges for precise, straight cuts and long lasting use. Stainless steel.... |
|
|
The Closer: The Complete Sixth Season $18.85 Kyra Sedgwick's Brenda Johnson is one of TV's most compelling, complicated, and likable heroines. Johnson makes The Closer much more than just the well-written, crisply paced police procedural that it is. The viewer feels invested in Johnson's neurotic private life, marriage, and family, as well as in her political ambitions at the Los Angeles Police Department and in her relationships with her co... |
|
|
Rizzoli & Isles: The Complete Second Season $27.99 ... |
|
|
Body of Proof: The Complete First Season $10.95 The always-welcome Dana Delany makes a vivaciously prickly return to television in this entertaining medical whodunit, which finds its groove after a fairly shaky start. Creator Christopher Murphey's premise draws inspiration from both Patricia Cornwell's Kay Scarpetta novels and the origin of Marvel comics' Dr. Strange: after injuring her hands in a car accident, an arrogant former neurosurgeon n... |
|
|
Midland GXT1000VP4 36-Mile 50-Channel FRS/GMRS Two-Way Radio (Pair) (Black/Silver) $50.09 This pair of Midland handheld GMRS radios allows you to keep in communication up to 36 miles apart during extreme outdoor activities. These feature-loaded radios include NOAA weather alert to automatically inform you of severe weather hazard information 24/7. Waterproof protection from splashing water and built-in SOS siren for emergencies. Channels (qty.): 50, Range (miles): 36, Privacy Codes (qt... |