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supremum and infimum

dwsmith

Well-known member
Feb 1, 2012
1,673
$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, where $a < b < c < d$

This questioned shouldn't be to difficult but would it be best to multiply out?

And how is the $a < b < c < d$ going to affect it?
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, where $a < b < c < d$

This questioned shouldn't be to difficult but would it be best to multiply out?

And how is the $a < b < c < d$ going to affect it?
What is asked?
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, where $a < b < c < d$

This questioned shouldn't be to difficult but would it be best to multiply out?

And how is the $a < b < c < d$ going to affect it?
Hi dwsmith, :)

It's clear that the set \(S\) contains elements \(a<x<b\) or \(c<x<d\). Otherwise, \((x - a)(x - b)(x - c)(x - d) >0\). That is,

\[S=\{x : a<x<b \mbox{ or }c<x<d\}=(a,b)\cup(c,d)\]

Now I suppose it is obvious as to what is the supremum and what is the infimum. Isn't? :)

Kind Regards,
Sudharaka.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Hi dwsmith, :)

It's clear that the set \(S\) contains elements \(a<x<b\) or \(c<x<d\). Otherwise, \((x - a)(x - b)(x - c)(x - d) >0\). That is,

\[S=\{x : a<x<b \mbox{ or }c<x<d\}=(a,b)\cup(c,d)\]

Now I suppose it is obvious as to what is the supremum and what is the infimum. Isn't? :)

Kind Regards,
Sudharaka.
$\text{inf} \ S = a + c$ and $\text{sup} \ S = b + d$
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
$\text{inf} \ S = a + c$ and $\text{sup} \ S = b + d$
\(a+c\) may not be a lower bound and \(b+d\) may not be an upper bound. A simple example to contradict your supremum and infimum would be, \(a=1,b=2,c=3,d=4\). Then,

\[S=(1,2)\cup(3,4)\]

Now it is clear that, \(1+3=4\) is not a lower bound of \(S\). \(2+4=6\) although an upper bound for this example is not the least upper bound.

The simplest way to think about this would be to draw the two intervals \((a,c)\) and \((b,d)\) on a real line(Note that, \(a<b<c<d\)) and see what are the upper bounds and lower bounds of \(S\).

Kind Regards,
Sudharaka.