Thursday, April 17, 2008

Set Theory and Binary Code

Imagine we have a set of numbers: {1, 2, 3}. How many subsets can we make out of that? We can take none of the numbers as a set, this is known as the null set and is represented by this symbol (). Then we can take each single number as set. Since there are three numbers, we will get three sets. Another three sets can be produced if we take two of the numbers like so: {1,2}{1,3}{2,3} and finally if we take all the numbers as a subset of the original set. This is known as the power set and has more things in it than the original set or we can say that it has a higher "cardinality". That brings our total to 8 subsets from the original set. What if we take a set of all the numbers from 1 to infinity. What we would get is an infinite set. The hebrew letter Aleph (א) with subscript zero is used to denote this. We can take the power set of that and have a set with a greater cardinality. This would be א with a subscript of 1. We can infinitely take the power set and get infinities larger than the previous infinity.

Why does 123 equal to 123? How can such a number exist when we only have 10 numbers. We use place value in base 10 to conjure up such numbers as 1596, 45 ,100, etc. Why base 10 though... according to Mr. K it's probably because we have 10 fingers but what if we use another base like 5. If we use this base 5, what would 123 be? Well the 3 would be the ones place, 2 would be the fives place and 1 would be the twenty-fives place. 123 in base 5 would actually be 33 back in base 10. Unlike us, computers calculate numbers in base 2 also known as the binary numeral system. There are only two numbers in binary, 0 and 1. Any number can be written in base 2 or any base for that matter. Even though we can do all these magnificent things with computers, at the heart of the matter, all the computer really does is just decide if a switch is on or off as in a computer chip there are little switches.. This is all the computer is actually able to do and understand.

After all that business, we took a crack at solving two logarithm problems. Eventually it all came down to just applying the logarithm laws. Following that, we looked the properties of a exponential function. The domain, range, x-intercepts, y-intercepts, concavity, if it was increasing or decreasing and the asymptote(s).

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