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Title: Restriction
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Series: Advent of Mathematical Symbols
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YouTube-Title: Restriction
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Bright video: https://youtu.be/48Af_zAZvuI
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Dark video: https://youtu.be/qF_srEONdzo
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: aoms09_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,900 –> 00:00:04,606 The mathematical symbol of today is the restriction for maps.
2 00:00:04,871 –> 00:00:07,014 Denoted with such a vertical line.
3 00:00:07,743 –> 00:00:11,729 and usually on the left hand side you find a map f
4 00:00:11,800 –> 00:00:14,478 and at the index you find a set A.
5 00:00:15,514 –> 00:00:20,900 Now since on the fundamental level in mathematics we often deal with maps and sets
6 00:00:21,229 –> 00:00:24,545 this restriction operator occurs very often.
7 00:00:25,414 –> 00:00:30,504 Indeed this whole thing here just denotes a new map, when we already have a map f.
8 00:00:31,129 –> 00:00:36,296 More precisely you would start with a map f, defined on a set X.
9 00:00:37,343 –> 00:00:40,892 and the codomain we call Y. So we map into Y.
10 00:00:41,943 –> 00:00:44,012 However now we want to change the domain.
11 00:00:44,212 –> 00:00:46,782 So we look at a subset A of X.
12 00:00:47,629 –> 00:00:54,104 This means now, that this f restricted to A should be a map that is only defined on the set A.
13 00:00:54,800 –> 00:00:58,107 So the domain is A, but we still map into Y.
14 00:00:59,029 –> 00:01:06,941 and now the only question we have is: how can we define this new map, we could call the restriction of f to A.
15 00:01:07,657 –> 00:01:12,934 Indeed the definition is very simple, because it should do the same as f itself.
16 00:01:13,986 –> 00:01:19,805 This means if we put in an x, we get the same result no matter which function we use.
17 00:01:20,614 –> 00:01:27,943 However you should note here for our new map, we are only allowed to use points x that come from the set A.
18 00:01:28,757 –> 00:01:34,521 So you see the reason for introducing this new map is not for calculating the values.
19 00:01:35,371 –> 00:01:41,902 Rather we use it if we want to work with the map as an object, but now defined on a smaller set.
20 00:01:42,571 –> 00:01:47,098 Therefore you see this symbol very often, when you learn mathematics.
21 00:01:47,843 –> 00:01:52,429 So i hope that this was helpful and that i see you in the next video.
22 00:01:52,430 –> 00:01:53,114 Bye!
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Quiz Content
Q1: Let $f: A \rightarrow B$ be a map. What is the domain of $f$?
A1: $A$
A2: $B$
A3: $A \cup B$
A4: $A \cap B$
Q2: Let $f: A \rightarrow B$ be a map and $C \subseteq A$. What is the domain of $f|_C$?
A1: $A$
A2: $B$
A3: $C$
A4: $A \cap B$
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Last update: 2024-11
