# Supremum and infimum of a set

1. Aug 30, 2006

### Valhalla

let $$B = \{x\in\mathbb{R} : sinx \geq 0 \}$$

find the supremum and infimum of this set.

Ok well, since it is periodic I guess the point would be to note that the set will repeat ever $$2\pi$$

So then if we consider just between 0 and $$2\pi$$

supremum = $$\pi$$
infimum = 0

if we consider all $$\mathbb{R}$$

here is where I'm confused. The supremum would just be the $$N\pi$$ when N is an odd integer. Should I just state the function is periodic it will repeat between 0 and $$2\pi$$

Last edited: Aug 30, 2006
2. Aug 30, 2006

### quasar987

Proceed methodically from the definition. M is a supremum of B if it is the smallest superior bound. But is B even bounded superiorly?

3. Aug 30, 2006

### Valhalla

Ok so I think I see what your saying. The set will not be bounded above or below except by plus or minus infinity b/c the function is periodic. I can always find a larger number in the reals that satifies sin(x) greater than or equal to 0. Therefore the set would have a supremum or positive infinity and a infimum of negative infinity.

Oh, I made typo in the original problem $$x\in\mathbb{R}_e$$

4. Aug 30, 2006

### quasar987

That is the idea, yeah. You'd have to write a few equations though for it to be considered a proof. You'd have to show rigorously that given any number in B, there is always another number in B that is superior(resp. inferior) to it.

(What is $\mathbb{R}_e$??)

5. Aug 30, 2006

### Valhalla

Our professor stated that $$\mathbb{R}_e$$ is the extended reals which contains plus and minus infinity. This course is an analysis for electrical engineers we get a crash course in a little bit of set theory then a bunch about complex functions with linear algebra of complex functions. We don't have a textbook for this course and the professors only written some of the course notes so I'm kind of flying blind on what is going on here. Thanks for the help!