What is the supremum and infimum of the set B = {x\in\mathbb{R} : sinx \geq 0}?

[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

 

[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?

 

[Valhalla]

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

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

 

[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??)

 

[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!