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| − | The beauty of induction is that it allows a theorem to be proven true where an infinite number of cases exist without exploring each case individually. Induction is analogous to an infinite row of dominoes with each domino standing on its end. If you want to make all the dominoes fall, you can either:
| + | Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: |
| − | #push on the first one, wait to see what happens, and then check each domino afterwards (which may take a long time if there's an infinite number of dominoes!)
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| − | #or you can ''prove'' that if any domino falls, then it will cause the domino after it to fall. (i.e. if the first one falls then the second one will fall, and if the second one falls then the third one will fall, etc.)
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| − | Induction, essentially, is the methodology outlined in point 2.
| + | :<blockquote>Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). |
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| − | ==Parts of Induction==
| + | :::— Concrete Mathematics, page 3 margins. </blockquote> |
| − | Induction is composed of three parts:
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| − | #The Base Case (in the domino analogy, this shows the first domino will fall)
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| − | #The Induction Hypothesis (in the domino analogy, we assume that a particular domino will fall)
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| − | #The Inductive Step (in the domino analogy, we prove that the domino we assume will fall will cause the next domino to fall)
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| − | ==Weak Induction== | + | A proof by induction consists of two cases. The first, the base case (or basis), proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the truth of the statement for all natural numbers n ≥ N. |
| − | Weak induction is used to show that a given property holds for all members of a countable [[#inductiveSet|inductive set]], this usually is used for the set of natural numbers.
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| − | Weak induction for proving a statement <math>P(n)</math> (that depends on <math>n</math>) relies on two steps:
| + | The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs. |
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| − | *<math>P(n)</math> is true for a certain base step. Usually the base case is <math>n=1</math> or <math>n=0</math>
| + | Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values. |
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| − | *<math>P(k)\to P(k+1)</math>. That is, if <math>P(k)</math> is true, then <math>P(k+1)</math> is true.
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| − | If these two properties hold, one may induce that the property holds for all elements in the set in question. Returning to the example, if you are sure that you called your neighbor, and you knew that everyone who was called in turn called his/her neighbor, then you would be guaranteed that everyone on the block had been called (assuming you had a linear block, or that it curved around nicely).
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| − | ===Example 1===
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| − | The first example of a proof by induction is always 'the sum of the first n terms:'
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| − | :'''Theorem 2.4.1.''' For any fixed <math>n\in \mathbb N,</math> <math>\sum_{i=1}^{n}{i}=\frac{n(n+1)}{2}</math>
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| − | ;Proof
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| − | *Base step: <math>1=\frac{1\cdot2}{2}</math> , therefore the base case holds.
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| − | *Inductive step: Assume that <math>\sum_{i=0}^ni=\frac{n(n+1)}{2}</math> . Consider <math>\sum_{i=1}^{n+1}i</math> .
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| − | :<math>\begin{align}
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| − | \sum_{i=1}^{n+1}i&=\sum_{i=1}^ni+(n+1)=\frac{n(n+1)}{2}+n+1\\
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| − | &=\left(\frac{n}{2}+1\right)(n+1)\\
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| − | &=\frac{(n+1)(n+2)}{2}\\
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| − | &=\frac{(n+1)\big((n+1)+1\big)}{2}
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| − | \end{align}</math>
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| − | So the inductive case holds. Now by induction we see that the theorem is true.
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| − | ===Example 2===
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| − | Proposition: Prove that <math>4~|~f(n)=5^n+8n+3 </math> for <math> n\in\N</math>
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| − | : Note our parameter, <math>\text{for }n\in\N</math> This means it wants us to prove that it's true for all values of <math>n</math> which belong to the set (<math>\in</math>) of positive integers (<math>\N</math>).
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| − | Basis case:
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| − | : <math>\begin{align}&\text{Let }n=1\\&f(1)=5^1+8(1)+3\implies f(1)=16\\&\therefore 4~|~f(1)\end{align}</math>
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| − | Assumption: Now we let <math>n=k</math> where <math>k</math> is a general positive integer and we assume that <math>4\ |\ f(k)</math>
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| − | : Remember <math>f(k)=5^k+8k+3</math>
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| − | Induction: Now we want to prove that the <math>k+1^{th}</math> term is also divisible by 4
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| − | : Hence <math>\text{let }n=k+1\implies f(k+1)=5^{k+1}+8(k+1)+3</math>
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| − | : This is where our assumption comes in, if <math>4~|~f(k)</math>then 4 must also divide <math>f(k+1)-f(k)</math>
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| − | : So: <math>f(k+1)-f(k)=5^{k+1}+8(k+1)+3-(5^k+8k+3)</math>
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| − | :: <math>\begin{align}&f(k+1)-f(k)=5(5^k)-5^k+8\\&f(k+1)-f(k)=4(5^k)+8\\&\therefore~f(k+1)-f(k)=4(5^k+2)\end{align}</math>
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| − | : Now we've shown <math>4~|~\bigl(f(k+1)-f(k)\bigr)</math> and thus <math>4~|~f(k+1)</math> it implies <math>4~|~f(n)</math> because you have successfully shown that 4 divides <math>f(n)</math>, where <math>n</math> is a general, positive integer (<math>k</math>) and also the consecutive term after the general term (<math>k+1</math>)
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| − | Conclusion:
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| − | :: <math>\begin{align}&4~|~f(k)\implies4~|~f(k+1)\\&\therefore~4~|~f(n),\forall n\in\Z^+\end{align}</math>
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| − | If <math> 4 </math> divides <math> f(k) </math>, then it implies that <math> 4 </math> also divides <math> f(k+1) </math>. Therefore, since <math> 4 </math> divides <math> f(1) </math>, <math> 4 </math> divides <math> f(n) </math> for all <math> n\in\N </math>.
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Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:
Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).
- — Concrete Mathematics, page 3 margins.
A proof by induction consists of two cases. The first, the base case (or basis), proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the truth of the statement for all natural numbers n ≥ N.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs.
Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values.
