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Complete induction is equivalent to ordinary mathematical induction as described above, in the sense that a proof by one method can be transformed into a proof by the other. Suppose there is a proof of P ( n ) {\displaystyle P(n)} by complete induction.
The argument above makes the implicit assumption that the set of + horses has the size at least 3, [3] so that the two proper subsets of horses to which the induction assumption is applied would necessarily share a common element. This is not true at the first step of induction, i.e., when + =.
In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. [ 15 ]
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. [1]
Inductive reasoning refers to a variety of methods of reasoning in which broad generalizations or principles are derived from a set of observations. [1] [2] Unlike deductive reasoning (such as mathematical induction), where the conclusion is certain, given the premises are correct, inductive reasoning produces conclusions that are at best probable, given the evidence provided.
We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0) Each equation follows by definition [A1]; the first with a + b, the second with b. Now, for the induction. We assume the induction hypothesis, namely we assume that for some ...
Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor). Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.
Among the applications of the product rule is a proof that = when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). The proof is by mathematical induction on the exponent n.