This is the part that confuses me. Should I be saying something to the effect:
For any a\inA there exists a unique b\inB and a unique c\inC such that f(a)=b and g(b)=c. It follows that g(f(a))=c. Thus (g o f)^(-1)(c)=a
"Since c is arbitrary, it suffices to prove that (g∘f)−1(c)=a as well. How can you do that?"
I don't quiet understand your question, or the statement before it.
Homework Statement
Let f:A->B and g:B->C be invertible mappings. Show (g o f)^-1 = f^-1 o g^-1.
Homework Equations
A mapping is invertible iff it is bijective
The Attempt at a Solution
I understand why these are equivalent statements; however, I can't figure out the wording of the...
Homework Statement
Prove the pigeonhole principle directly. so basically |Nk-{x}|=|Nk-1| if k>1 is an integer and x belongs to the set of naturals.
Homework Equations
The Attempt at a Solution
I have no idea even where to begin.
I can never remember how to expand a summation in to form: \sumnk=1(22). Thats just a recent example. I can't remember the expansion form any sort of summation really except when it has a defined upper bound.
Sorry this took so long. Originally I was looking at how you did it not the question and what the answer should be. You forgot a negative from d/dx(1/x) but I'm pretty sure you can't do it that way. As far as I know you can approach this straight on or using a ln trick. I'll show both...
Well the only mistake I see in your first work is that you forgot a negative. and for the second problem break it down to a more simple problem. It may become more work to resimplify it later but you aren't wrong to do so.
You can take it and make it into this d/dx((x-1)(1/2)/(x2(x-4))(1/2))...
You were right in the fact that a direct proof would be much to hard for this problem, but a contradiction is hard to come up with. See if you can prove it by contrapositive Suppose f(x) doesn't equal zero, however according to your givens it has to be greater than or equal to zero , then it...
Ok this isn't so urgent any more but my teacher loves to just stare blankly at me like I'm stupid when I ask questions so I'd like to continue here with anyone who will help. I got to |(Y[n]*L-M*L+M*L-X[n]*M)/X[n]*L|<epsilon in which i then factored |(L*(Y[n]-M)-M*(X[n]-L))/X[n]*L|<epsilon. I...
Could you clarify this at all. What i sounds like you are saying to me is that I should be supposing the consequent but that is uber taboo. I mean I could suppose the my antecedent and then bind |Y[n]/X[n]-M/L|<epsilon but I'd have to get there algebraically first.
Homework Statement
Prove that if X[n]->L, X[n] doesn't equal 0 for all n, L doesn't equal 0 and Y[n]->M, then (Y[n]/X[n])->(M/L).
Homework Equations
They only give you the squeeze theroem and that if X[n] converges then it's limit is unique. O and the definition. A sequence of reals has...