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Change the problem in .NET Implementation barcode data matrix in .NET Change the problem

Change the problem using barcode integration for visual .net control to generate, create data matrix barcodes image in visual .net applications. Visual Studio 2010 The third method to get visual .net data matrix barcodes started is to change the problem. What we do is specialize and generalize.

These two methods are so important that they are given their own chapter, 33. For the moment a good example of changing the problem is the general version of the Division Lemma on page 199. One assumption was that y was a non-zero integer.

However, the bulk of the proof involved proving a different statement, the one where y is a positive integer. Once that part of the proof was done a separate, and shorter, argument. Getting to a higher level 245 dealt with the negative Data Matrix ECC200 for .NET y case. This method can be used again and again.

If the statement says integers it may be easier to use the natural numbers. More generally, we can restrict to a smaller class of objects and investigate those. The point is that, by changing the question, we at least can do something.

. Getting to a higher level Getting started is of co ECC200 for .NET urse very important. But how do you go beyond merely answering questions and get to truly thinking like a mathematician Let s see some techniques.

. Reverse the question construct your own examples A real mathematician inv Visual Studio .NET ECC200 ents their own examples. Here, I don t only mean nding examples of de nitions and theorems as suggested in earlier chapters, useful as that is.

An example of what I mean is given in calculus. Finding the maxima and minima of functions is a basic exercise in calculus: you are given a function and have to nd its critical points. Now the technique for creating examples is: Reverse that question.

Suppose that you are given that the maxima and minima are at certain points. Can you construct a function which has these maxima and minima You will learn lots about maxima and minima from doing this. Obviously one could just guess a function and work with that.

But a higher level of understanding is to be gained from creating an example with some pre-de ned restrictions, e.g. a maximum at point x = 2 with value f (x) = 5 and minima at x = 2 and x = 7 with f ( 2) = 20 and f (7) = 3.

For another example of this reverse-the-question method, consider conics from elementary geometry. The standard question is Here is a conic, calculate its centre, vertices, eccentricity and so on. Reverse this question.

Given conditions involving the centre, vertices and eccentricity create a conic with these properties.. Ask What happens if . . . Good mathematicians like .net framework datamatrix 2d barcode to ask What happens if . .

. For example, what happens if I drop that assumption We have seen above that this can help solve problems. It also allows us to explore the limits of a subject.

We can nd out why de nitions and theorems are the way they are. As another example, mathematical objects are often sets with some extra conditions. At a very simple level we can say that a nite set is one with a nite number of elements but there are much more complicated examples such as groups.

(A group is a set with a way of multiplying elements of the set. The multiplication has to satisfy certain properties.) Now, for sets A and B we can take their product A B.

We can ask, if A and B have a certain property, then does A B For example, suppose that A and B are nite sets.. CHAP T E R 32 Putting it all together Is A B a nite set In .net framework barcode data matrix this case, yes it is. If A and B are in nite, then is A B in nite If A and B are groups, then is A B a group And so on.

The idea is that we are always asking questions to create new concepts..
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