induction : Dictionary / Wörterbuch (BEOLINGUS, TU Chemnitz)

Proverbs, aphorisms, quotations (English)	by Linux fortune

Every program has at least one bug and can be shortened by at least one instruction -- from which, by induction, one can deduce that every program can be reduced to one instruction which doesn't work.
Several students were asked to prove that all odd integers are prime. The first student to try to do this was a math student. "Hmmm... Well, 1 is prime, 3 is prime, 5 is prime, and by induction, we have that all the odd integers are prime." The second student to try was a man of physics who commented, "I'm not sure of the validity of your proof, but I think I'll try to prove it by experiment." He continues, "Well, 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is... uh, 9 is... uh, 9 is an experimental error, 11 is prime, 13 is prime... Well, it seems that you're right." The third student to try it was the engineering student, who responded, "Well, to be honest, actually, I'm not sure of your answer either. Let's see... 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is... uh, 9 is... well, if you approximate, 9 is prime, 11 is prime, 13 is prime... Well, it does seem right." Not to be outdone, the computer science student comes along and says "Well, you two sort've got the right idea, but you'll end up taking too long! I've just whipped up a program to REALLY go and prove it." He goes over to his terminal and runs his program. Reading the output on the screen he says, "1 is prime, 1 is prime, 1 is prime, 1 is prime..."
Excitement and danger await your induction to tracer duty! As a tracer, you must rid the computer networks of slimy, criminal data thieves. They are tricky and the action gets tough, so watch out! Utilizing all your skills, you'll either get your man or you'll get burned! -- advertising for the computer game "Tracers"
Conjecture: All odd numbers are prime. Mathematician's Proof: 3 is prime. 5 is prime. 7 is prime. By induction, all odd numbers are prime. Physicist's Proof: 3 is prime. 5 is prime. 7 is prime. 9 is experimental error. 11 is prime. 13 is prime ... Engineer's Proof: 3 is prime. 5 is prime. 7 is prime. 9 is prime. 11 is prime. 13 is prime ... Computer Scientists's Proof: 3 is prime. 3 is prime. 3 is prime. 3 is prime...
Lemma: All horses are the same color. Proof (by induction): Case n = 1: In a set with only one horse, it is obvious that all horses in that set are the same color. Case n = k: Suppose you have a set of k+1 horses. Pull one of these horses out of the set, so that you have k horses. Suppose that all of these horses are the same color. Now put back the horse that you took out, and pull out a different one. Suppose that all of the k horses now in the set are the same color. Then the set of k+1 horses are all the same color. We have k true => k+1 true; therefore all horses are the same color. Theorem: All horses have an infinite number of legs. Proof (by intimidation): Everyone would agree that all horses have an even number of legs. It is also well-known that horses have forelegs in front and two legs in back. 4 + 2 = 6 legs, which is certainly an odd number of legs for a horse to have! Now the only number that is both even and odd is infinity; therefore all horses have an infinite number of legs. However, suppose that there is a horse somewhere that does not have an infinite number of legs. Well, that would be a horse of a different color; and by the Lemma, it doesn't exist.
Theorem: All positive integers are equal. Proof: Sufficient to show that for any two positive integers, A and B, A = B. Further, it is sufficient to show that for all N > 0, if A and B (positive integers) satisfy (MAX(A, B) = N) then A = B. Proceed by induction: If N = 1, then A and B, being positive integers, must both be 1. So A = B. Assume that the theorem is true for some value k. Take A and B with MAX(A, B) = k+1. Then MAX((A-1), (B-1)) = k. And hence (A-1) = (B-1). Consequently, A = B.
Mid-Twenties Breakdown: A period of mental collapse occurring in one's twenties, often caused by an inability to function outside of school or structured environments coupled with a realization of one's essential aloneness in the world. Often marks induction into the ritual of pharmaceutical usage. -- Douglas Coupland, "Generation X: Tales for an Accelerated Culture"
Actually, the probability is 100% that the elevator will be going in the right direction. Proof by induction: N=1. Trivially true, since both you and the elevator only have one floor to go to. Assume true for N, prove for N+1: If you are on any of the first N floors, then it is true by the induction hypothesis. If you are on the N+1st floor, then both you and the elevator have only one choice, namely down. Therefore, it is true for all N+1 floors. QED.
Proof techniques #1: Proof by Induction. This technique is used on equations with "_n" in them. Induction techniques are very popular, even the military used them. SAMPLE: Proof of induction without proof of induction. We know it's true for _n equal to 1. Now assume that it's true for every natural number less than _n. _N is arbitrary, so we can take _n as large as we want. If _n is sufficiently large, the case of _n+1 is trivially equivalent, so the only important _n are _n less than _n. We can take _n = _n (from above), so it's true for _n+1 because it's just about _n. QED. (QED translates from the Latin as "So what?")