Reference materials: computer science, mathematics
Another one of my blog: review what it is to prove that the discrete series ①
Well-ordering principle
Definition: non-empty set of non-negative integer elements must be minimized.
Yes, you read right, the well-ordering principle is so obvious. However, the well-ordering principle is one of the most important discrete mathematics principles.
Well-ordered prove
Well-ordered proved to be a proven method of use of the well-ordering principle. Reductio ad absurdum is proven and well-ordered linked, if used well-ordered proof, it will use the reductio ad absurdum.
We look at an example:
Example: to demonstrate any non-negative integer n, 1 + 2 + 3 + ..... + n = n (n + 1) / 2
Through this sample item, I think you can feel the basic role of the well-ordering theorem. We read on:
Well-ordered prove template
Use well-ordering theorem proof "of all the n- \ (\ in \) . N, the p-(the n-) established" (generally used to prove a well-ordered prove Whatever the problem
- Use contradiction, the definition set C is set to true counterexample P
- The well-ordering principle, there must be a minimum element n- \ (\ in \) C
- ---- conflict resolution is typically P (n) is smaller than the presence of an element n, or C is true. It depends in part on the specific task of proof.
- Concluded, C must be the empty set, i.e., the absence of counter-example.
Well-ordered collection
If there is a minimal element of a set of arbitrary non-empty set, we call this set is well-ordered.
(This is not very important, we will not start the detailed
Some exercises
Personally think that in order to understand and use well-ordered prove, is the need to exercise more from the summarized refining, here are some well-ordered prove exercises:
Some exercises
to sum up
Well-ordering principle is "fundamental theorem of thinking," well-ordered and proved to be a mathematical proof method well-ordering principle. Generally used as proof "of all the n- \ (\ in \) N, the p-(the n-) set up a" kind of problem.