- #1

JasonJo

- 429

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(1) let f: [a,b] --> R be an integrable function. Consider a sequence (Pn) of tagged partitions with ||Pn|| -> 0. Prove that int over [a,b] of f(x) = lim n-> infinity of S(f, (Pn)).

- i stated what it means for a function to be integrable, using the episolon-delta definition, but i can't seem to find the step i need.

(2)

(a) Let f: [0,1] -> R be defined by:

f(x) = 1 if x=1/n for n a natural number

= 0 otherwise

is f integrable? if so find int of f(x) over [0,1]

(b) same question for g: [0,1] -> R

g(x)=n if x=1/n for some n a natural number

= 0 otherwise

(3) prove that a homomorphism from a field to a ring with more than one element must be an isomorphism.

- i got that kerT, where T is our homomorphism, is trivial. i just don't get the onto proof yet. why is T onto? i tried assuming it's not onto and trying to derive a contradiction that kerT is not trivial.

check it out:

suppose T is not onto, then there exists an element r in the ring R such that there does not exist f in the field such that T(f) = r. however, there exists r1 + r2 = r and f1 and f2 such that T(f1)=r1 and T(f2)=r2, then T(f1+f2)=r1+r2=r, which is a contradiction. but this doesn't seem to be right

(4) prove that Q[x]/<x^2-2> is ring isomorphic to Q[sqrt(2)].

- i can't find a proper isomorphism. really stuck

thanks guys! last homeworks of the semester!

- i stated what it means for a function to be integrable, using the episolon-delta definition, but i can't seem to find the step i need.

(2)

(a) Let f: [0,1] -> R be defined by:

f(x) = 1 if x=1/n for n a natural number

= 0 otherwise

is f integrable? if so find int of f(x) over [0,1]

(b) same question for g: [0,1] -> R

g(x)=n if x=1/n for some n a natural number

= 0 otherwise

(3) prove that a homomorphism from a field to a ring with more than one element must be an isomorphism.

- i got that kerT, where T is our homomorphism, is trivial. i just don't get the onto proof yet. why is T onto? i tried assuming it's not onto and trying to derive a contradiction that kerT is not trivial.

check it out:

suppose T is not onto, then there exists an element r in the ring R such that there does not exist f in the field such that T(f) = r. however, there exists r1 + r2 = r and f1 and f2 such that T(f1)=r1 and T(f2)=r2, then T(f1+f2)=r1+r2=r, which is a contradiction. but this doesn't seem to be right

(4) prove that Q[x]/<x^2-2> is ring isomorphic to Q[sqrt(2)].

- i can't find a proper isomorphism. really stuck

thanks guys! last homeworks of the semester!

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