Supremum and Infimum of $S$: $a < b < c < d$

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In summary, the set $S$ is defined as all values of $x$ where the expression $(x-a)(x-b)(x-c)(x-d)$ is less than $0$. This means that $S$ contains elements between $a$ and $b$ or between $c$ and $d$, and does not include $a$, $b$, $c$, or $d$. The supremum of $S$ is $b+d$ and the infimum is $a+c$. However, these may not always be the actual upper and lower bounds of $S$, as demonstrated by the example $a=1,b=2,c=3,d=4$. It is best to visualize the two intervals $(a,c)$
  • #1
Dustinsfl
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$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, 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?
 
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  • #2
dwsmith said:
$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, 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?
 
  • #3
dwsmith said:
$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, 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.
 
  • #4
Sudharaka said:
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 $\text{sup} \ S = b + d$
 
  • #5
dwsmith said:
$\text{inf} \ S = a + c$ 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.
 

Related to Supremum and Infimum of $S$: $a < b < c < d$

1. What is the definition of supremum and infimum?

The supremum of a set $S$ is the least upper bound of $S$, meaning it is the smallest number that is greater than or equal to every element in $S$. The infimum of $S$ is the greatest lower bound of $S$, meaning it is the largest number that is less than or equal to every element in $S$.

2. How are supremum and infimum related to each other?

The supremum and infimum are related in that they are both bounds of a set $S$. The supremum is the smallest upper bound, while the infimum is the largest lower bound. In other words, the supremum is always greater than or equal to the infimum.

3. What is the significance of $a < b < c < d$ in the context of supremum and infimum?

In this context, $a$ represents the infimum of the set $S$, while $d$ represents the supremum of $S$. The numbers $b$ and $c$ are elements of $S$ and fall between the infimum and supremum. This notation helps to illustrate the relationship between the infimum, supremum, and elements of $S$.

4. Can a set have more than one supremum or infimum?

No, a set can only have one supremum and one infimum. However, a set can have multiple elements that are equal to the supremum or infimum. For example, if $S = \{1,2,3\}$, then both $2$ and $3$ are equal to the supremum and infimum of $S$.

5. How can we determine the supremum and infimum of a given set?

To determine the supremum and infimum of a set $S$, we first need to find the upper and lower bounds of $S$. Then, we can compare these bounds to the elements of $S$ to find the smallest upper bound and largest lower bound, which will be the supremum and infimum, respectively. In some cases, the supremum or infimum may not exist if there is no upper or lower bound.

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