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The if and only if nature of mathematical de nitions in .NET Encoder DataMatrix in .NET The if and only if nature of mathematical de nitions




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The if and only if nature of mathematical de nitions using barcode drawer for visual .net control to generate, create data matrix 2d barcode image in visual .net applications. Visual Studio 2005/2008/2010 Overview Giving de nitions is on .net framework ECC200 e area where mathematicians are imprecise. In a de nition with an if , what is intended is an if and only if .

The only if part of a de nition is considered to be such an obvious part that it is omitted. For example, the de nition above A positive integer n is a square number if n = x 2 for some integer x should be read as an if and only if statement: A positive integer n is a square number if and only if n = x 2 for some integer x. In other words the number is called square if the condition is true but, more than that, only if the condition is true.

There are no square numbers that do not satisfy the condition. This is an important point to bear in mind when reading mathematical de nitions. It is the only time that an if can be read as an if and only if .

Do not do this when reading theorems for example.. How to read a de nition 105 How to read a de nition Observe Obviously, given a de n VS .NET barcode data matrix ition one should observe precisely the conditions given. Bear in mind that we are not allowed to read in anything extra.

Note that in an example all the conditions need to be true. Not just some. For example, in the de nition of squarey-twinney above, the prime needs to be a twin prime and needs to be squarey.

So, for example, if a prime is a twin and not squarey, then it is not squarey-twinney.. What are we dealing with The rst task is to ide .NET Data Matrix 2d barcode ntify what we are dealing with. Is it something we already know well with an extra property For example, a twin prime is just a prime p with an additional property: p 2 or p + 2 is a prime.

We can ask other questions. Is it similar or different to a de nition already known Is it analogous to something else Is it a de nition we know plus a new condition For example, a proper subset of X is a subset with the additional property that it is not equal to X..

What examples of this de nition exist Given a de nition, we n Visual Studio .NET Data Matrix 2d barcode eed to ask if such an object exists. Admittedly, it is unlikely that you will be given a de nition of an object that does not exist! The point is to improve understanding by initially being sceptical.

If such objects do exist, how common are they Is the object unique Is there a nite number An in nite number Returning to our examples beginning on page 103, obviously, even and odd numbers exist. There is obviously an in nite number of nite sets. So, there is a plentiful supply of them, and it is easy to construct examples.

What about twin primes Well, 5 and 7 are twin. So are 41 and 43. But these may be the only ones.

If you look at a list of primes less than 1000 you will see that there are more than just these initial examples. Then we can ask, is there an in nite number of twins Interestingly, nobody knows! For another example recall squarey numbers above. A natural number is called squarey if its digits are the last digits of its square.

Do examples exist for this Obviously 52 = 25 and 25 ends in a 5, so 5 is squarey. But are there any others Well, experimentation shows that 762 = 5776, so 76 is also squarey. Looking at the numbers from 1 to 100, we nd that 1, 5, 6, 25, and 76 are all squarey.

Thus only ve out of the rst hundred numbers have this property. It is easy to see that this property should in some sense get rarer as our numbers get bigger. This gives us some idea of the notion of squarey.

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