Supremum and Infimum of a subset of R

[Rubik]

I do not understand how to calculate the sup[tex]\Omega[/tex] and inf[tex]\Omega[/tex] of a subset of R. So for example calculating the sup and inf of [tex]\Omega[/tex] = (1,7)U[8,[tex]\infty[/tex]) and the answer is no sup and inf = 1. I do not know how to get these values?

 

[Fredrik]

Let S be a set of real numbers. sup S exists if and only if S is bounded from above, i.e. if there exists a real number M such that for all x in S, x≤M. If this is the case, M is said to be an upper bound of S, and sup S is defined as the least (i.e. smallest) upper bound. (We don't have to prove that sup S exists, because its existence is part of the definition of the real numbers). This means that if M is an upper bound of S, then sup S≤M.

inf S is defined similarly, as the greatest lower bound of S. If S=(1,7)⋃[8,∞), then clearly, S isn't bounded from above, and is bounded from below by any number ≤1. Since inf S is the greatest lower bound of S, this means that 1≤inf S. So now we just need to prove that inf S≤1. This is almost obvious, but let's do it right: For every x>1, there exists a y in S such that y<x. This means that no real number >1 can be a lower bound of S. inf S is a lower bound, so it must be ≤1.

 

[Rubik]

Okay so if I then have the set {x is an element of R : |3x + 7| > 1}

How do I get the sup of this set? Does that mean there is no sup?

 

[TylerH]

Solve |3x + 7| > 1 for the explicit ranges of x. Then see if there is a lowest number higher than all of the x in that range.

 

[Fredrik]

Okay so if I then have the set {x is an element of R : |3x + 7| > 1}

How do I get the sup of this set? Does that mean there is no sup?

I solved your first problem completely. If you want help with more, you need to show us your work so far, and where you're stuck. TylerH told you where you should start.