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Expert: - 5/6/2009


Question
Hello again Scott,
Would you proof the following for me?

This is a long one and I believe I got it correct but I would appreciate a set of fresh eyes (yours!)

Let f: A arrow B be a function and S be a subset of A, T be a subset of A.  Does f (S - T) = f(S) - f(T)? If so prove it, otherwise, give a counter example.

PF. Let y belong f(s-t).  By definition of image y=f(x)., where x belongs to (S-T).  To prove equality of sets, we must first prove the sets are subsets of each other.  We proceed as follows:
1) f(S-T) subset f(S) - f(T).  Assume y belongs to f(S-T).  By def. of set-theoretic difference, x belongs to S and x does not belong to T, and by definition of Domain and Codomain it follows f(x) belongs to f(S) and f(x) does not belong to f(T), so y belongs to f(S) and y does not belong to f(T). Thus, by def. o set-theoretic difference, y belongs f(S)- f(T). Therefore, f(S-T) subset f(S) - f(T).
2) f(S) - f(T)is subset of f(S-T). Assume y belongs f(S)- f(T).  Then, by set-theoretic diff., y belongs f(S) and y does not belong f(T) and by def. of Domain and Codomain x belongs S and x does not belong to T.  Thus, f(S) - f(T)is subset of f(S-T).
Therefore, the sets are equal by definition of equality of subsets.

I greatly appreciate it.

Evelyn

Answer
I'm thinking about that one...


                                  and thinking ...

       and looking ...

        and thinking  ...
                                        and understanding ...
  
            and analyzing ...

It looks good the way it is.
If I have a different opinion after reviewing it again tonight,
I'll send an update to my answer.

Hopefully it will form a union between the following two sets:
1) understanding the answer  - and -
2) knowing the answer is right.

They are close together and I believe that the answer is OK,
but I'll analyze it some more.

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Scott A Wilson

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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My clients have been students at OSU, people nearby, friends with math questions, and several people every day on the PC, and you're probably make one more.

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