Mathematical induction: Difference between revisions

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imported>Michael Hardy
imported>Michael Hardy
(I've entirely replaced the first paragraph, which was fairly vague. That paragraph's definition of "induction hypothesis" was completely wrong.)
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In [[mathematics]], an '''inductive proof''' is a [[proof]] by cases, applicable whenever the problem can be divided into discrete, enumerable propositionsInductive proofs consist first prove a base proposition <math>P_0</math>, and then prove an inductive hypothesis <math>\forall i > 0, P_i\implies P_{i+1}</math>.  [[modus ponens]] is then used to extend the proof over the entire domain of the problem.
'''Mathematical induction''' is a technique for proving a proposition that can be considered a conjunction of an infinite sequence of special cases.  Despite the name, it is a deductive, rather than inductive, method, in the sense in which those terms are usually used in [[logic]].
 
In any proof by mathematical induction, one proves that each member of an infinite sequence of propositions holds by reasoning as follows:
 
* The first case is true; and
* If any case is true then so is the next.
 
Suppose one calls the propositions ''P''<sub>1</sub>, ''P''<sub>2</sub>, ''P''<sub>3</sub>, ... Then the proof consists of two steps:
 
* The proof that ''P''<sub>1</sub> is true; and
* The proof that if ''P''<sub>''k''</sub> is true, the so is ''P''<sub>''k''+1</sub>, regardless of the value of ''k''.
 
The first step is called the "basis"The second is the "inductive step".  In the inductive step one assumes the truth of ''P''<sub>''k''</sub> and proves ''P''<sub>''k''+1</sub>.  The proposition ''P''<sub>''k''</sub> is called the "induction hypothesis".


[[Category: Mathematics Workgroup]] [[Category: CZ_Live]]
[[Category: Mathematics Workgroup]] [[Category: CZ_Live]]
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== Example ==
== Example ==


Proposition: A [[tree (Graph Theory)|tree]] in [[graph theory]] has [[euler characterist|Euler Characteristic]] of -1.
Proposition: A [[tree (graph theory)|tree]] in [[graph theory]] has [[euler characterist|Euler characteristic]] &minus;1.


Proof: By induction,
Proof: By induction,

Revision as of 11:38, 13 July 2007

Mathematical induction is a technique for proving a proposition that can be considered a conjunction of an infinite sequence of special cases. Despite the name, it is a deductive, rather than inductive, method, in the sense in which those terms are usually used in logic.

In any proof by mathematical induction, one proves that each member of an infinite sequence of propositions holds by reasoning as follows:

  • The first case is true; and
  • If any case is true then so is the next.

Suppose one calls the propositions P1, P2, P3, ... . Then the proof consists of two steps:

  • The proof that P1 is true; and
  • The proof that if Pk is true, the so is Pk+1, regardless of the value of k.

The first step is called the "basis". The second is the "inductive step". In the inductive step one assumes the truth of Pk and proves Pk+1. The proposition Pk is called the "induction hypothesis".

Example

Proposition: A tree in graph theory has Euler characteristic −1.

Proof: By induction,

Base proposition: the trivial tree ---a single vertex without edges---has Euler characteristic .

Inductive Hypothesis: For a tree , and any extension single vertex extension of that tree , show that .

If , then adding one vertex and one edge to this graph would yield:

.

Since all trees can be constructed in this manner from the trivial tree, it must be the case that all trees have Euler Characteristic -1. QED.