Is {z: z doesn't equal 1 + i} an open and/or a closed set?

Question

Also,
which of the following sets are open and/or closed?

{z: (Re(z))^2 - (Im(z))2 less than or equal to 1}

{z: -Pi/4 < arg z < Pi/4}

Thanks

Answer 1

Here are some ways to think about the question. I'll just address your first example.

Call the set S. ( all points in the complex plane except for 1 + i)

1) is it true for any point w in S, you can draw a disk surrounding w with the entire disk inside of S? If so, S is open. On the other hand, if you can find even one point w IN S for which every disk centered at w includes points outside of S, then S is not open.

2) Is it true that every sequence of points in S has its limit points IN S? If so, then S is closed. Or can you find a sequence in S that has a limit point outside of S? If so, then S is not closed.

And for both parts, what makes this problem not so hard is that there is only ONE point outside of S. so focus your attention there!

Answer 2

A={z: z doesn't equal 1 + i} is an open set.

If z0 in A, z0<>i+1 (not equal), then a=|z0-(1+i)| > 0, then there the set |z-z0| < a is contained in A, then A is open.

{z: (Re(z))^2 - (Im(z))2 less than or equal to 1} is closed

{z: -Pi/4 < arg z < Pi/4} is open

Answer 3

In general, if you have strict inequalities or "not equals" relationships, you'll have open sets. If you have <= or >= you'll have open sets.

Very few sets are both open and closed.

Answer 4

at the same time as including professional numerals you in difficulty-free words positioned the whole: *bear in suggestions you may in difficulty-free words upload them if the have a similar pronumeral (see eg 3) eg. z+z+z+z = 4z eg. a+a = 2a eg. m+m+h+h = 2m + 2h wish this helps