Being humans, this entry will be particularly sobering: math can be used to determine when our species will die out. It’s a fair assumption that most of the readers of this article are human beings. This logical paradox caused a complete reformation of Set Theory, one of the most important branches of math today. But we now we have to put it back…and so on. But wait…now it DOES contain itself, so naturally we have to take it out. But what about Russell’s set itself? It doesn’t contain itself, so surely it should be included as well. The set of all fruit doesn’t contain itself (the jury’s still out on whether it contains tomatoes), so it can be included in Russell’s set, along with many others. Russell decided to get meta about things and described a set that contained all those sets which do not contain themselves. In 1901 famous mathematician Bertrand Russell made quite a splash when he realized that this way of thinking had a fatal flaw: namely, not anything can be made into a set. Additionally, and this is important, sets can contain other sets (like the set of all sets in the preceding sentence). The thinking of the time was that anything could be turned into a set: The set of all types of fruit and the set of all US Presidents were both completely valid. Basically, a set is a collection of objects. At the turn of the 20th century, a lot people were entranced by a new branch of math called Set Theory (which we’ll cover a bit later in this list). Taking the square root of both sides gives. The right side can be written as a single fraction, with common denominator 4 a 2. The left side is now a perfect square because In order for these equations to be true,Īdding this number to equation (1) makes. To "complete the square" is to find some number k so thatįor another number y. The quadratic equation is now in a form in which completing the square can be done. īy a (which is allowed because a is non-zero), gives: Prove that x = -b ± ( √b² - 4ac ) / 2a from ax²+bx+c=0. When the contradiction appears in the proof, there is usually an X made with 4 lines instead of 2 placed next to that line. When proving a theorem by way of contradiction, it is important to note that in the beginning of the proof. That is, if one of the results of the theorem is assumed to be false, then the proof does not work. Proof by contradiction is a way of proving a mathematical theorem by showing that if the statement is false, there is a problem with the logic of the proof. Then for n= n 0+1, 2 can be rewritten 2( n 0+1) + 2 Next, assume that for some n= n 0 the statement is true. Proof: First, the statement can be written "for all natural numbers n, 2 =n(n+1)įirst, for n=1, 2 =2(1)=1(1+1), so this is true. Prove that for all natural numbers n, 2(1+2+3+.+ n-1+ n)= n( n+1) Induction shows that it is always true, precisely because it's true for whatever comes after any given number. And since it's true for 3, it must be true for 4, etc. And since it's true for 2, it must be true for 3. Since it's true for some beginning case (usually n=1), then it's true for the next one ( n=2). Once that is shown, then it means that for any value of n that is picked, the next one is true. Show that the statement is true for the next value, n 0+1. Assume that for some value n = n 0 the statement is true and has all of the properties listed in the statement. Prove that the statement is true for some beginning case.ģ. State that the proof will be by induction, and state which variable will be used in the induction step.Ģ.
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