Bounded sets in Higher Dimensions

Bounded Sets

Consider the set $\mathbb {N} =\{1,2,3,...\}$. As you might imagine, there is no "largest" element in this set. If we were to claim that some element $u\in \mathbb {N}$ was the largest element in $\mathbb {N}$, then we note that $u+1\in \mathbb {N}$ and $u<u+1$. There is a least element in this set though, namely $1$. We can say that every element $n\in \mathbb {N}$ is such that $1\leq n$. We could alternatively say that every element $n\in \mathbb {N}$ is such that $0\leq n$ or $-3234\leq n$. We will now formally define this sort of idea.

Definition: Let $S\subseteq \mathbb {R}$ where $S\neq \emptyset$. We say that $S$ is bounded above by $M$ if for all $x\in \mathbb {S}$, $x\leq M$. We say that $S$ is bounded below by $m$ if for all $x\in \mathbb {S}$, $m\leq x$. We say that $S$ is bounded if it is both bounded above and bounded below, and we say that $S$ is unbounded if it is not both bounded above and below.

From this definition, we can intuitively say that the set of natural numbers $\mathbb {N}$ is bounded below by any number $m\leq 1$. However, we note that $\mathbb {N}$ is not bounded above since such as $M$ does not exist. Therefore in general we say that the set of natural numbers is not bounded.

Another example is the set $S:=\{x\in \mathbb {R} :-2\leq x\leq 2\}$. As we can guess, this set $S$ is bounded below by $m=-2$ and bounded above by $M=2$. Therefore we say that $S$ is bounded.

Sometimes a set might not be bounded above and might also not be bounded below. For example consider the set of integers $\mathbb {Z}$ which is clearly a subset of $\mathbb {R}$. This set is not bounded below and not bounded above.

We will now look at some theorems regarding bounded sets.

Theorem 1: If $A\subseteq B$ and $B$ is a bounded set then $A$ is a bounded set. Further, if $A\subseteq B$ and $A$ is an unbounded set then $B$ is an unbounded set.

• Proof: Let $A\subseteq B$.

• For the first part of the proof, let $B$ be a bounded set and let $m$ be any lower bound to $B$ and let $M$ be any upper bound to $B$. Then if $x\in A$, then $x\in B$, but $\forall x\in B$ we have that $m\leq x\leq M$, and so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall x \in A} , $m\leq x\leq M$ so $A$ is a bounded set.

• For the second part of the proof, let $A$ be an unbounded set and suppose instead that $B$ is a bounded set. Let $m$ be any lower bound to $B$ and let $M$ be any upper bound to $B$. Then if $x\in A$, then $x\in B$, but $\forall x\in B$ we have that $m\leq x\leq M$, and so $\forall x\in A$, $m\leq x\leq M$, so $A$ is bounded, but that contradicts the fact that $A$ is an unbounded set, so our assumption that $B$ was bounded is false. Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is an unbounded set. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

Note that in the proof of Theorem 1, we could have omitted the second part of the proof since it is the contrapositive of the first part.

Bounded Subsets in Euclidean Space

Definition: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq \mathbb{R}^n} . The set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is said to be Bounded if there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} \in \mathbb{R}^n} and a positive real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq B(\mathbf{x}, r)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is said to be Unbounded otherwise.

In other words, a subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} is bounded if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is the subset of some ball in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} .

In Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} , a set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq \mathbb{R}^2} that is bounded might look something like:

Meanwhile, a set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} that is unbounded might look something like:

For a more concrete example, consider the subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = [0, 1) \times [0, 1) \subseteq \mathbb{R}^2} . Then the ball Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(\mathbf{0}, 2)} is such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq B(\mathbf{0}, 2)} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is bounded. However, the subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = \{ (x, y) \in \mathbb{R}^2 : x \geq 0, y \geq 0 \} \subseteq \mathbb{R}^2} is unbounded. To prove this, set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} = \mathbf{0}} . Suppose that there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq B(\mathbf{0}, r)} . now consider the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{y} = (r+1, r+1) \in S} . Then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \| \mathbf{r} - \mathbf{0} \| = \sqrt{(r+1)^2 + (r+1)^2} = \sqrt{2} \mid r + 1 \mid > r \end{align}}

Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (r + 1, r + 1) \not \in B(\mathbf{0}, r)} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} , but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (r+1, r+1) \in S} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \not \subseteq B(\mathbf{0}, r)} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is unbounded.

Licensing

Content obtained and/or adapted from:

• Bounded Sets, mathonline.wikidot.com under a CC BY-SA license
• Bounded Subsets in Euclidean Space, mathonline.wikidot.com under a CC BY-SA license