Using a two column proof in StatementReason format prove the

Using a two column proof (in Statement-Reason format), prove the following:

i) If S is a subset of R(real numbers), then every interior point of S is in S, and every exterior point of S is in R - S.

ii) Suppose S is a subset of R(real numbers). Prove:

iia) S and R - S have the same boundary.

iib) The interior of S is the same as the exterior of R - S.

iic) The exterior of S is the same as the interior of R - S. Any help is greatly appreciated.

1) If S is inside R, then everypoint inside S is in S. If it lies outside S, is in the area of R-S.

2) a) S has a boundry. Its shared by the ring R exclusing S. So, R-S has the same boundry

. So, S and R-S have same boundry.

b)Outside R-S ring is S. So, anything apart from R-S( ext. of R-S) lives in S.

So The interior of S is the same as the exterior of R - S.

c) S is part of R. Of you go out of S though, you reach the area not including S but in R. So, you are in R-S range.

So, exterior of S(outside) is the same as the interior of R - S.