Counterexamples in .NET Integrating datamatrix 2d barcode in .NET Counterexamples

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Counterexamples using barcode integration for visual studio .net control to generate, create barcode data matrix image in visual studio .net applications. VB.NET A counterexample is a special type of example. De nition 12.1 An example which shows .net framework data matrix barcodes that a statement is false is called a counterexample. The counter part of the word comes from the fact that we are countering, in the sense of rejecting or rebutting, the truth of a statement.

. Examples 12.2 (i) Consider All prime Visual Studio .NET datamatrix 2d barcode s are odd. The number 2 is a counterexample to this statement.

(ii) The statement If I am Churchill, then I am English was used as a counterexample to A implies B is equivalent to not(A) implies not(B). . How to create examples and counterexamples (iii) Consider Let p a barcode data matrix for .NET nd q be real numbers. If p/q Q, then p Q and q Q.

This can seem quite reasonable. However, let p = /3 and q = /2 and we see that p/q = 2/3. Thus p and q are real numbers such that p/q Q and yet both p and q are not in Q.

Thus p = /3 and q = /2 provides a counterexample to the statement.. Remark 12.3 A statement of the form ECC200 for .NET If , then cannot be shown to be true by taking a particular example of where the hypotheses are true and the conclusion is true. However, if we can nd just one example for which the statement is not true, then the statement is not a theorem.

Therefore, to show a statement is wrong you need only one counterexample. One and one only is needed..

Exercise 12.4 Find a counterexample to The square of a real number is positive. How to create examples and counterexamples A lot of the internal d datamatrix 2d barcode for .NET ialogue of a mathematician is about examples and counterexamples, yet they are often neglected in books and lecture courses. As I said earlier it is often very dif cult to construct examples and counterexamples, but it is where the true mathematician has a chance to shine.

Thus your rst reaction to being given an example is to nd another example. But that is not the only time. You should cultivate a sceptical mind.

Hence, if someone makes a statement, then your reaction should be to disbelieve them and attempt to nd an example that shows the statement is false. Even if their statement is true, the mental workout that this process gives is bene cial. It also helps develop a feel for the statement.

Note that constantly doing this in real-life situations can lose you friends people tend to get upset if you are constantly nding fault with what they say! Given any statement try to nd a counterexample.. Example 12.5 A letter to a newspaper barcode data matrix for .NET stated that time travel is impossible because of logic: If time travel were possible, then one would meet lots of people from the future. Reading this I tried to nd many reasons why it was not true.

Try it before reading the following. I had some ideas why it might be wrong. Maybe time travel only allows us to travel forward in time (by amounts larger than we do already!).

Maybe time travellers are not allowed to communicate with us. Maybe time travel has a range, you can t travel back more than a year and time travel is still a number of years away (and time travel machines can t be transported)..

CHAP T E R 12 Examples and counterexamples In the case of mathemat Data Matrix 2d barcode for .NET ics how does one produce these examples One way is to have a good stock of examples in your head ready to be used at a moment s notice, for example, the empty set or Z. The other way is practice.

That is, the more practice you have in creating examples and counterexamples, then the better and quicker you become at creating them. Just as there is no sure- re method for problem-solving, there is no magic formula for creating examples. One of the reasons that standard texts avoid examples and counterexamples is that this lack of a magic formula makes it hard to teach their use.

Creating examples and counterexamples is initially frustrating as rst attempts are often not very good. It is important to persevere as the ability is key to excellence in mathematics..

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