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bar code for vb CHAP T E R 20 Techniques of proof I: Direct method in .NET Generation data matrix barcodes in .NET CHAP T E R 20 Techniques of proof I: Direct method




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CHAP T E R 20 Techniques of proof I: Direct method use none none integrated toprint none in nonevb.net to create barcode Choose the sid none none e with the most complicated expression and reduce it to the other side; (ii) x y = 0; (iii) x y and x y; (iv) x = z and y = z. To prove X Y : assume x X and proceed to show that x Y . To prove that two sets X and Y are equal: (i) use a string of equalities, or (ii) prove that X Y and Y X.

To prove that A B show A = B and B = A . Summary of summary: (i) x = y is equivalent to x y = 0, (ii) x = y is equivalent to x y and y x, (iii) X = Y is equivalent to Y X and Y X, (iv) A B is equivalent to A = B and B = A ..

Microsoft Windows Official Website Some common mistakes To err is huma none for none n. To really mess up takes a computer. Anon.

The direct method of proof is probably the most basic and will be used in the other methods. Despite this there are a couple of pitfalls that are easy to fall into. Two of the most common are assuming what had to be proved and incorrect use of equivalence.

We shall investigate these in this chapter. And since it would be nice to gather common mistakes together in one handy chapter rather than having them distributed throughout the book some other mistakes are included. Also brought in is an explanation of why we can t divide by zero a mistake you probably already know about but may not have been given a reason why.

I have seen all these errors made and, like most mathematicians, have made them myself.. Don t assume what had to be proved Probably the m none none ost common mistake in proofs is assuming what had to be proved. Suppose that we had to prove statement P . If we assume it is true, then it is not surprising that we can deduce it is true; P = P would seem to be very obviously true.

Another error in this vein is that P is assumed to be true and this is used to deduce something that is true and so it is concluded that P is true. This is of course an incorrect argument. (It says P = Q, but Q is true, so P is true.

) As an example, consider the following statement: If a and b are real numbers, then a 2 + b2 2ab. A fallacious proof is: We have a 2 + b2 2ab = a 2 2ab + b2 0 = (a b)2 0. The last inequality is true as the square of a number is always non-negative, so a 2 + b2 2ab.

. CHAP T E R 21 Some common mistakes The error here none none is that the conclusion (in the statement to be proved) has been assumed (i.e. that a 2 + b2 2ab) and has led to something we know is true.

However, we cannot conclude that a statement is true just because it implies a known truth. Consider the true statement 1 = 1 = 1 = 1 discussed on page 65. The 1 = 1 part implies 1 = 1 , which is true but obviously we cannot conclude from 1 = 1 that 1 = 1 .

The real proof is just a reverse of the argument; begin with (a b)2 0, something we know is true, and proceed to the conclusion we want by reversing the implication signs above. Try it and see!. It is ok to as sume what has to be proved when nding a proof!. Now for the re none for none ally confusing piece of advice. I have just said that you should not assume what had to be proved when proving statements. However, when it comes to solving problems, i.

e. nding the proof, it is ok to assume what had to be proved. We saw examples of this repeatedly in the previous chapter.

This strategy will often unlock the problem and allow us to create a proof. In the above example we saw how assuming the conclusion a 2 + b2 2ab led us to (a b)2 0, something we know is true. Fortunately, in this case it might not be true in other cases we can reverse the implication arrows and go from (a b)2 0 to a 2 + b2 2ab.

Thus, when solving problems it is ok as an exploratory tool to assume what had to be proved, just to see where it leads, but when writing a polished version it is not ok..
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