# supremum and infimum

#### dwsmith

##### Well-known member
$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, 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
$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, 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?

#### Sudharaka

##### Well-known member
MHB Math Helper
$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, 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
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
$\text{inf} \ S = a + c$ 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.