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A proof by induction consists of two cases. The first, the base case, proves the statement for = without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case =, then it must also hold for the next case = +.
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] This ...
We already saw in the base case that the rule ("all horses have the same color") was valid for =. The inductive step proved here implies that since the rule is valid for n = 1 {\displaystyle n=1} , it must also be valid for n = 2 {\displaystyle n=2} , which in turn implies that the rule is valid for n = 3 {\displaystyle n=3} and so on.
Base case may refer to: Base case (recursion) , the terminating scenario in recursion that does not use recursion to produce an answer Base case (induction) , the basis in mathematical induction, showing that a statement holds for the lowest possible value of n
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.
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Trim off the base and any damaged areas, then use as directed. Tip For an even more tender texture, peel off the tough membrane of the celery ribs before using.