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To generalize is to restrict oneself to the essentials…


How much is the power "n raised to zero"? We already saw that 2 raised to m is nothing more than the amount of parts of the set m. Now, in the case of power 2 raised to zero, our m It is the empty set. So our question is how many parts the empty set has. Of course the only part of the empty set is the empty part. We easily deduce that 2 raised to zero can only be 1 which is the number of parts (or subsets) of the empty set.

The generalization process in mathematics is a fundamental process. We could say that what the mathematician does all the time is trying to generalize a truth that is already known to him, or just out of curiosity, even if he does not know if a statement is true, the mathematician may ask if it allows a "generalization". It is very easy to give examples of interesting and important generalizations. Right here, right now, we are facing an interesting and important example: as we find that the powers of 2 are simply the result of counting the parts of the exponent sets. m, it is irresistible to ask if the powers, say, no raised to m, would also not be the result of object counts in certain situations. Remember that the natural numbers themselves can be thought of as representations of the count results: zero (or empty set) is the result of counting the sets belonging to the empty set, 1 is the result of counting the sets belonging to the set. {Æ}, and so on. You could say: "natural numbers are the counting process itself". We fully agree with this view which, by the way, is very elegant.

But back to today's theme, could we "generalize" the idea that a power of 2 counts the parts of its exponent m and then ask if a base power no and exponent m Wouldn't that be an interesting count? If so, we need to know "what" count? The area of ​​mathematics that deals with this kind of problem, namely the object counting problem in certain situations, is the Combinatorial. It is a fascinating area, just like any other area of ​​mathematics. It has developed remarkably in recent decades, as has all of mathematics. We are living a time of great progress in mathematics. To give you an idea, you can already find in good bookstores, such as the excellent Livraria Cultura de São Paulo, books about the Human Genome Combinatorics, a very important area that will surely make great progress in the coming months.

But let's not lose the thread: how is the idea of ​​counting behind the powers of 2 generalized? To generalize is to restrict oneself to the essential. So what we need to do is see what is essential in the count behind a power of 2. Remember that the transformations of the set m in the set {0, 1} = 2 gave us the idea of ​​separation of the set m in two packages. One package was that of the sets of m that were associated with 0 and the remaining package was the package of the leftover sets. The essence, therefore, of a function or transformation of the set m in set 2 is the idea of ​​"color wrapping", that is, for each set y (color y) of the set no we formed the "color package y"of sets x in m. When no is 2 this simply gives us the count of the parts of m.

So that was the hypothesis that base powers no and exponent m they are also results of interesting counts that are more general than counting the parts of the set. m. The powers n raised to m simply count in how many ways we can form packets of n colors with the sets of m. Try it yourself: make sure that this combinatorial interpretation of the natural powers of natural numbers works even in the cases of 2 cube and 3 cube. Remember that it is allowed to form empty packages to respect the amount of color required. Of course you can now answer without hesitation: any natural power of exponent 0 is 1!

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