- #1
scorpion990
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Hey everybody... I have a few quick questions concerning sets and functions for the experts... I've been trained in applied mathematics, so I'm not really used to this kind of formalism.
1. Can somebody look at my "proposed proof" of this elementary theorem for me? I have a feeling that it is correct, although it's not giving the satisfaction of truly understanding the basics. In the very least, I'm skipping steps that are otherwise important to include in order to see exactly what's going on. I won't feel right continuing on without having the absolute basics down cold. Here it is:
http://img205.imageshack.us/img205/5286/proofqy7.png
2. I'm also trying to prove the theorem labeled "2" in the previous image. I understand the logic behind it, but I'm having trouble writing it down in a rigorous manner. At best, I can describe it by an example.
Let:
B = {1,2,3}
C = {2,4,6,8}
f: X --> Y
f(x) = 2x
The function f maps the set X into the set Y. The function is one-to-one because there exists a unique x for a given by. It is not "onto" because there exists elements in Y which do not correlate to the set X under f.
f inverse of the set C is the set B. However, the element "8" in C does not correlate to B. Applying f on the set B returns C' = {2, 4, 6}, which is a subset of the original set C.
I have absolutely no idea how to convert that into logical steps involving set theory and the definitions of functions. The book that I'm using gives no example proofs =/
Here are a few quick questions that I'm having trouble on. Any insight is appreciated.
3. "How many subsets are there of the set {1,2,3,...,m}? How many maps of this set into itself? How many maps of this set onto itself?
The first answer is clearly "an infinite number of subsets". I'm not quite so sure about the latter two questions. I don't really understand the difference between the two, so that makes it even more complicated. For the first, if they assume that X is the set of integers, and so is Y, then there would be one function that maps one into the second: f: X-->y. f(x) = x
4. This question kind of goes along with the third one. I'm really confused, though.
a. "How many functions are there from the nonempty set S into the empty set?"
I guess... one? f: X-->Y such that for all x, y = undefined
b. "How many functions are there from the empty set into an arbitrary set S?"
An infinite number of functions? Any defined function with the domain of "no elements" will map into an arbitrary set S.
Any help with any of the questions posed above are appreciated. Thanks.
1. Can somebody look at my "proposed proof" of this elementary theorem for me? I have a feeling that it is correct, although it's not giving the satisfaction of truly understanding the basics. In the very least, I'm skipping steps that are otherwise important to include in order to see exactly what's going on. I won't feel right continuing on without having the absolute basics down cold. Here it is:
http://img205.imageshack.us/img205/5286/proofqy7.png
2. I'm also trying to prove the theorem labeled "2" in the previous image. I understand the logic behind it, but I'm having trouble writing it down in a rigorous manner. At best, I can describe it by an example.
Let:
B = {1,2,3}
C = {2,4,6,8}
f: X --> Y
f(x) = 2x
The function f maps the set X into the set Y. The function is one-to-one because there exists a unique x for a given by. It is not "onto" because there exists elements in Y which do not correlate to the set X under f.
f inverse of the set C is the set B. However, the element "8" in C does not correlate to B. Applying f on the set B returns C' = {2, 4, 6}, which is a subset of the original set C.
I have absolutely no idea how to convert that into logical steps involving set theory and the definitions of functions. The book that I'm using gives no example proofs =/
Here are a few quick questions that I'm having trouble on. Any insight is appreciated.
3. "How many subsets are there of the set {1,2,3,...,m}? How many maps of this set into itself? How many maps of this set onto itself?
The first answer is clearly "an infinite number of subsets". I'm not quite so sure about the latter two questions. I don't really understand the difference between the two, so that makes it even more complicated. For the first, if they assume that X is the set of integers, and so is Y, then there would be one function that maps one into the second: f: X-->y. f(x) = x
4. This question kind of goes along with the third one. I'm really confused, though.
a. "How many functions are there from the nonempty set S into the empty set?"
I guess... one? f: X-->Y such that for all x, y = undefined
b. "How many functions are there from the empty set into an arbitrary set S?"
An infinite number of functions? Any defined function with the domain of "no elements" will map into an arbitrary set S.
Any help with any of the questions posed above are appreciated. Thanks.
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