4. Let E CRd, and for each nE N, define On as the set On z d(x, E) 1/n) Recall that d(x, E) inf{| - el : e E E}. Prove that for each n e N, On is open

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Q: b. Let E = [0, 1] ~ P. Then µ*E = 1, and if A N [0, 1] is a measur- able set, then m*(A N E) = mE.

A: Given: E=0,1~P To prove: m*(A∩E)=mE

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Q: a. Write a proof for if f:x->Y and g:Y-Z are 1 to 1 then the composition fog is 1-1 as well.   b.…

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Q: 26. Let E be the set of all even integers, and let O be the set of all odd integers. Let X = = {n =…

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Q: 7. Show that if S x T is a closed subset of R?, then S and T are closed subsets of R.

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Q: 7. Let E and 0 the set of even and odd integers, respectively. Prove that {E, O} partitions Z.

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Q: 3. Let f: A→ Band g: B → Cdenote two functions. Show that if gofis 1-1, then fis 1-1. D JCI

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Q: 2.12 let A and B bounded sets For which there is an ε >o such that la-blze for all ac A, be B .…

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Q: 22. Let A be a bounded subset of R and let sup A = B. Which of the following is/are not an attribute…

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Q: Represent in the Argand Diagram all such z for which |z − 5i| > 8

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Q: 7. a) Let N G, H ≤ G. Let NH = = {nh: n € N, h E H}. Show that NH ≤ G. b) Let H ≤ G and [G:H]=2.…

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Q: 6. (a) Fix n, m € Z4. For each i e N,, define A; = {f : Nm → N„|i ¢ im(f)}. Prove (by constructing a…

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