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Mathematical Induction
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Mathematical Induction is the "dominoes theory" of proof. It is often used when you want to prove something is true for every whole number, but don't have time to prove all the infinite cases one by one. It has two parts: line up the dominoes, and knock over the first one. Lining up the dominoes means you prove that if the statement is true for one number, it's true for the next number. Knocking over the first domino is just proving that it works for the first number (usually one.) Mathematicians call these the Induction Step and the Base Case.
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Proof by Mathematical Induction presents the Automated Deduction community with some very challenging research problems. The aim of this one day workshop is to create an informal forum in which hot-topics and emerging techniques can be presented and discussed. The workshop continues and focuses the efforts of previous workshops held in conjunction with AAAI'93, CADE'94, CADE'96, CADE'97, CADE'98.
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Proof by Mathematical Induction presents the Automated Deduction community with some very challenging research problems. The aim of this one day workshop is to create an informal forum in which hot-topics and emerging techniques can be presented and discussed.
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The principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms. However, it can be proved in some logical systems. For instance, it can be proved if one assumes:
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Mathematical induction is not only useful for proving algebraic identities. It can be used any time you have a recursive relationship--one where the current case depends on one or more of the previous cases. See the second example below for a geometric application of induction. Here’s a more general way to think about induction:
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The principle of mathematical induction stated below describes the above (previous) ladder idea in the algebraic shorthand notation favored in mathematics. The last part of this chapter will not make sense to you if you are not familiar with this shorthand notation. If this is the case, you may skip this description of mathematical induction. If you read it, and you find that you do not understand it, you could return to it later after you have seen the following chapters on algebra. They explain the use of shorthand in algebra.
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Mathematical Induction