Math Is Fun Forum

http://www.mathisfunforum.com/viewtopic.php?id=10891

I solved this problem in the above link using vectors and basic algebra.

Maybe im missing something...

We need to find g such that g(g(x)) = f(x) and f(x) = x + 1

So then let g(x) = x + 1/2 and:

g(g(x)) = (x + 1/2) + 1/2 = x + 1 = f(x)

You have a given g(x) function, when it says to find a function g, does it mean to find a new function?

Hello Jane,

Was there anything wrong in my written proof?

Thanks.

Remember that the power set is a set itself and anytime you want to prove that a set is equal to another set you have to show two things, that is you need to prove that one set is a subset of the other and vice versa. If you have, for example, sets A and B, and you want to show that A = B then you need to show that A is a subset of B and B is a subset of A. That proves that A = B.

So we want to prove:

P (A ∩ B) = P (A) ∩ P (B)

So we take arbitrary x in P(A ∩ B). Since x is in P(A ∩ B) then we know that x is a subset of A ∩ B by definition of the power set. Since x is in A ∩ B then we know that x is in A and x is in B. Since x is in A and in B then it follows that x is in P(A) and x is in P(B). Hence x is in P (A) ∩ P (B). So we have showed that P (A ∩ B) is a subset of P (A) ∩ P (B).

So now we take arbitrary x in P (A) ∩ P (B). So it follows that x is in P(A) and x is in P(B). By the definition of the power set, it follows that x is a subset of A and x is a subset of B. Since x is a subset of A and B, then it follows that x is in A ∩ B. Since x is in A ∩ B, then it follows that x is in the power set of A ∩ B, that is x is in P(A ∩ B). So we have showed that P (A) ∩ P (B) is a subset of P(A ∩ B).

This completes the proof that P (A ∩ B) = P (A) ∩ P (B).

The class Axiomatic Set Theory is actually part of the Math Department, do some universities teach it within the Philosophy Department? I was surprised to hear your question since I never heard that. For the class we are going to be using:

Introduction to Set Theory (3rd edition) by Karel Hrbacek & Thomas Jech
http://www.amazon.com/Introduction-Revised-Expanded-Applied-Mathematics/dp/0824779150/ref=sr_1_1?ie=UTF8&s=books&qid=1225129736&sr=8-1

For algebra we going to be using:

“Advanced Modern Algebra” by J. Rotman
http://www.amazon.com/Advanced-Modern-Algebra-Joseph-Rotman/dp/0130878685/ref=sr_1_1?ie=UTF8&s=books&qid=1225130025&sr=1-1

and for analysis:

Real Analysis by John M. Howie.

Have any opinion on this books?

Where A and B are sets?

I never knew that sets had addition defined.

I just cant picture the addition of two sets? My mind keeps thinking addition means union.

what is {1,2,5} + {2,3,6,7}? Just so I could see how you add sets.

Oh, so I guess my professor meant:

But he wrote it like a normal plus sign and not the "exclusive" sign like you said. Are they interchangeable when talking about sets?

If R(A) = R(A^2) that means that the set that A maps into is the same set that A^2 maps into as well. But im not really sure I get what does R(A) = R(A^2) really imply? What does it really mean that the range of A is the same as the range of A^2?

I tried to play with it for a while but I couldnt derive anything meaningful out of it. What does it mean if y = A(R(A)) etc?

If something is not in R(A) then it is in N(A) which is why R(A) U N(A) = R^n, I understand that part. I feel im missing "something" that just makes all this "click" together.

As I said, I been thinking and pondering about this for a couple days now but with my limited mathematical knowledge, I havent been able to derive anything. But its always good to have a problem in your head your always thinking about, I guess.

The thing is that each side individually is not always true. You have to assume one side and proof it implies the other side. Vice versa.

Is that what you meant?

im half Spanish and half Cuban. Although the Cuban part dates back to Spain as well...so Im 100% Spanish? I dont know.