(b) Is the converse valid? Give a proof or counterexample.#7. (a) Suppose U and V are open subsets of R". Prove that U U V and UnV are open as well.(Recall that UUV= {x e R2 xe U or x e V} and Un V = {x e R" : x e U and x e V}.)(b) Suppose C and D are closed subsets of R". Prove that C UD and CnD are closed as well.8. Let S C R". We say a e S is an interior point of S if some neighborhood of a is contained in S.

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Asked Aug 13, 2019
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I need help for number 7. Thanks!

(b) Is the converse valid? Give a proof or counterexample.
#7. (a) Suppose U and V are open subsets of R". Prove that U U V and UnV are open as well.
(Recall that UUV= {x e R2 xe U or x e V} and Un V = {x e R" : x e U and x e V}.)
(b) Suppose C and D are closed subsets of R". Prove that C UD and CnD are closed as well.
8. Let S C R". We say a e S is an interior point of S if some neighborhood of a is contained in S.
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(b) Is the converse valid? Give a proof or counterexample. #7. (a) Suppose U and V are open subsets of R". Prove that U U V and UnV are open as well. (Recall that UUV= {x e R2 xe U or x e V} and Un V = {x e R" : x e U and x e V}.) (b) Suppose C and D are closed subsets of R". Prove that C UD and CnD are closed as well. 8. Let S C R". We say a e S is an interior point of S if some neighborhood of a is contained in S.

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Step 1

(a)

Let

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Step 2

Then by definition, either x belongs to U or x belongs to V.

Assume that x belongs to U,

Since, U is open, there is some r > 0 so that the open ball,

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Br(x)CU >Br(x)CUUV (by definition of union)

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Step 3

Thus, union of two open subsets...

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xEUnV

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