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What is asked?$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, 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,$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, 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?
$\text{inf} \ S = a + c$ and $\text{sup} \ S = b + d$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.
\(a+c\) 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,$\text{inf} \ S = a + c$ and $\text{sup} \ S = b + d$