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 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 . |
For example consider the numbers and . 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}
