# Infimum and Supremum

• Nov 15th 2011, 06:56 AM
Shizaru
Infimum and Supremum
I know what infimum and supremum are but I would like a bit of help with some notations "Find the infimum and supremum (if they exist) of:

1). $\displaystyle [1,3)$
2). $\displaystyle [1, \infty )$
3). $\displaystyle (-2,0) \cup (1,2)$
4). $\displaystyle \{1 - \frac{1}{n} : n \in \mathbb{N}\}$
5). $\displaystyle \{\frac{1}{2^{n}} - \frac{1}{3^{m}}: n, m \in \mathbb{N}\}$"

Now, this is where the notes we have begin and I don't know what it means when you have one [ bracket with a ) to close. So for the first one, is it just supremum = 3 and infimum = 1, or do the different brackets mean something special?

Here's my attempts:

1). 3, 1
2). none, 1
3). 2, -2
4). 1, 0

5). Should 0 be considered a natural number? I have not heard of a standard convention on this. Texts seem to go either way. This would change the result of this one and without knowing I don't know how to handle it.

* (As a side can anyone tell me the tex for the element of and natural, real, rational etc symbols?).
• Nov 15th 2011, 07:09 AM
Siron
Re: Infimum and Supremum
To find the supremum and infimum of 5) it can be useful to write some elements of the set $\displaystyle \{ \frac{1}{2^n}-\frac{1}{3^m}|m,n \in \mathbb{N}\}$ down and conclude something.

To write $\displaystyle \mathbb{R}$ with latex you have to use [ tex ] \mathbb{R} [ /tex ]
• Nov 15th 2011, 07:26 AM
Plato
Re: Infimum and Supremum
Quote:

Originally Posted by Shizaru
(As a side can anyone tell me the tex for the element of and natural, real, rational etc symbols?).

Here are a few:

$$\cup$$ gives $\displaystyle \cup$ is union
$$\cap$$ gives $\displaystyle \cap$ is intersection
$$\subseteq$$ gives $\displaystyle \subseteq$ is subset
$$x\in\mathbb{R}$$ gives $\displaystyle x\in\mathbb{R}$ is element
$$\emptyset$$ gives $\displaystyle \emptyset$ is emptyset
$$\wedge$$ gives $\displaystyle \wedge$ is or
$$\vee$$ gives $\displaystyle \vee$ is and
$$\sqrt{x+1}$$ gives $\displaystyle \sqrt{x+1}$
• Nov 15th 2011, 11:48 PM
Shizaru
Re: Infimum and Supremum
Quote:

Originally Posted by Siron
To find the supremum and infimum of 5) it can be useful to write some elements of the set $\displaystyle \{ \frac{1}{2^n}-\frac{1}{3^m}|m,n \in \mathbb{N}\}$ down and conclude something.

This is already my approach. What I ask is whether 0 should be considered a natural number as this would affect the result.

It could be 1 and -1 or 1/2 and -1/3, depending if the lowest natural number is 0 or 1 (respectively). I mean as n and m tend to infinity both fractions tend to zero so it depends on whether n, m start at 0 or 1 as this dictates both the largest positive and negative you can make.

Also as for the brackets, please, someone tell me what it means to have one square and one round bracket. There is nothing in my notes explaining what this means... it just leads with the problems.
• Nov 16th 2011, 02:24 AM
Plato
Re: Infimum and Supremum
Quote:

Originally Posted by Shizaru
This is already my approach. What I ask is whether 0 should be considered a natural number as this would affect the result.

There is absolutely no agreement on the answer to this question.
It strictly depends upon your textbook/instructor.
For me $\displaystyle 0\in\mathbb{N}.$

Quote:

Originally Posted by Shizaru
Also as for the brackets, please, someone tell me what it means to have one square and one round bracket. There is nothing in my notes explaining what this means... it just leads with the problems.

$\displaystyle \begin{gathered} x \in \left[ {a,b} \right] \Leftrightarrow \,a \leqslant x \leqslant b \hfill\\ x \in \left( {a,b} \right) \Leftrightarrow \,a < x < b \hfill \\ x \in [a,b) \Leftrightarrow \,a \leqslant x < b \hfill \\ x \in (a,b] \Leftrightarrow \,a < x \leqslant b \hfill \\ \end{gathered}$
• Nov 16th 2011, 02:41 AM
Shizaru
Re: Infimum and Supremum
Quote:

Originally Posted by Plato
$\displaystyle \begin{gathered} x \in \left[ {a,b} \right] \Leftrightarrow \,a \leqslant x \leqslant b \hfill\\ x \in \left( {a,b} \right) \Leftrightarrow \,a < x < b \hfill \\ x \in [a,b) \Leftrightarrow \,a \leqslant x < b \hfill \\ x \in (a,b] \Leftrightarrow \,a < x \leqslant b \hfill \\ \end{gathered}$

Thanks very much, this is straightforward but wasn't in our notes. Am I right in thinking it wouldn't affect the supremum and infimum?
• Nov 16th 2011, 05:13 AM
Deveno
Re: Infimum and Supremum
that is correct. the only difference between (a,b) and [a,b], is that inf([a,b]) and sup([a,b]) are IN [a,b], you could replace them by min, and max., whereas (a,b) has no minimum, nor maximum element. this is a specific property of the real numbers, for example:

sup(A), where $\displaystyle A = \{x \in \mathbb{Q} : x^2 - 2 < 0\}$, is not a rational number, even though every element of A is.

so if we were working with just rational numbers, sup(A) would not exist! which means that the rational numbers has at least one "hole" in it (actually, LOTS of holes).

the lowly "<" sign, although it seems quite harmless, actually forces us to look for more numbers than rationals, because we can write meaningful rational inequalities that have no solution.