Difference between revisions of "Neighborhoods in 𝐑"
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<h1 id="toc0">The Real Number Line</h1> | <h1 id="toc0">The Real Number Line</h1> | ||
<p>One way to represent the real numbers <math>\mathbb{R}</math> is on the real number line as depicted below.</p> | <p>One way to represent the real numbers <math>\mathbb{R}</math> is on the real number line as depicted below.</p> | ||
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[[File:Real number line for Algebra book.svg|Real number line]] | [[File:Real number line for Algebra book.svg|Real number line]] | ||
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<p>We will now state the important geometric representation of the absolute value with respect to the real number line.</p> | <p>We will now state the important geometric representation of the absolute value with respect to the real number line.</p> | ||
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<td><strong>Definition:</strong> Let <math>a</math> be a real number and let <math>\varepsilon > 0</math>. The <math>\varepsilon</math>-neighbourhood of the number <math>a</math> is the set denoted <math>V_{\varepsilon} (a) := \{ x \in \mathbb{R} : \mid x - a \mid < \varepsilon \}</math>. Alternatively we can define <math>V_{\varepsilon}(a) := \{x \in \mathbb{R} : a - \varepsilon < x < a + \varepsilon \}</math>. | <td><strong>Definition:</strong> Let <math>a</math> be a real number and let <math>\varepsilon > 0</math>. The <math>\varepsilon</math>-neighbourhood of the number <math>a</math> is the set denoted <math>V_{\varepsilon} (a) := \{ x \in \mathbb{R} : \mid x - a \mid < \varepsilon \}</math>. Alternatively we can define <math>V_{\varepsilon}(a) := \{x \in \mathbb{R} : a - \varepsilon < x < a + \varepsilon \}</math>. | ||
| − | [[File:Epsilon Umgebung.svg|<math>\varepsilon</math>-neighbourhood around a (<math>V_{\varepsilon}(a)</math>) expressed on the real number line]] | + | |
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| + | [[File:Epsilon Umgebung.svg|<math>\varepsilon</math>-neighbourhood around <math>a</math> (<math>V_{\varepsilon}(a)</math>) expressed on the real number line]] | ||
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</td> | </td> | ||
Revision as of 14:10, 20 October 2021
The Real Number Line
One way to represent the real numbers 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}} is on the real number line as depicted below.
We will now state the important geometric representation of the absolute value with respect to the real number line.
| Definition: If and are real numbers, then we say that the distance from to the origin is the absolute value of , . We say that the distance between and is the absolute value of their difference, namely 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 \mid a - b \mid} . |
For example consider the numbers 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 -2} 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 2} . There is a distance 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 4} in between these numbers because 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 \mid -2 - 2 \mid = \mid -4 \mid = 4} .
Epsilon Neighbourhood of a Real Number
| 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 a}
be a real number and 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 \varepsilon > 0}
. The 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 \varepsilon}
-neighbourhood of the 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 a}
is the set denoted 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 V_{\varepsilon} (a) := \{ x \in \mathbb{R} : \mid x - a \mid < \varepsilon \}}
. Alternatively we can define 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 V_{\varepsilon}(a) := \{x \in \mathbb{R} : a - \varepsilon < x < a + \varepsilon \}}
.
|
For example, 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 1} , and 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 \varepsilon_0 = 2} . 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 V_{\varepsilon_0} (1) = \{ x \in \mathbb{R} : \mid x - 1 \mid < 2 \} = (-1, 3)} .
We will now look at a simple theorem regarding the epsilon-neighbourhood of a real number.
| Theorem 1: 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 a} be a real number. 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 \forall \varepsilon > 0} , 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 x \in V_{\varepsilon} (a)} 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 x = a} . |
- Proof of Theorem 1: Suppose that for some 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 x} , 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 \varepsilon > 0} , 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 \mid x - a \mid < \varepsilon} . We know that 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 \mid x - a \mid = 0} if and only 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 x - a = 0} and 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 x = 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 \blacksquare}
