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Title: Set Brackets
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Series: Advent of Mathematical Symbols
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YouTube-Title: Set Brackets
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Bright video: https://youtu.be/ZxnJkGFJB48
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Dark video: https://youtu.be/FTzO3m1uzT0
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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: aoms11_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,700 –> 00:00:03,821 The mathematical symbol of today is the set bracket.
2 00:00:04,021 –> 00:00:06,849 Which we use in mathematics to construct new sets.
3 00:00:07,614 –> 00:00:13,564 For example you often see something like this, that after the set bracket we have a function f(x).
4 00:00:14,571 –> 00:00:18,232 and then on the right you see a vertical line or a colon.
5 00:00:18,571 –> 00:00:24,286 Moreover next to it you find something like: x is an element in a set “A”.
6 00:00:25,286 –> 00:00:30,526 Of course there could be more conditions than this, but in the end we have a closing set bracket.
7 00:00:31,900 –> 00:00:40,763 Now the meaning of this is that we apply the function f to all elements x from the set “A” and then we form a new set.
8 00:00:41,614 –> 00:00:45,741 and the elements of the new set are exactly these values.
9 00:00:46,386 –> 00:00:52,445 So you could read it as the set of the elements f(x), where x comes from “A”.
10 00:00:53,114 –> 00:00:56,742 Indeed this is a construction you see very often in mathematics.
11 00:00:57,186 –> 00:01:00,425 Therefore it might be helpful to look immediately at an example.
12 00:01:01,243 –> 00:01:07,606 So lets look at a set of the elements where the function f is given by (2 times x + 1).
13 00:01:08,729 –> 00:01:14,335 and here x should come from the set {0, 1, 2, 3}.
14 00:01:14,886 –> 00:01:20,331 So you see we construct a new set using this set here and that function.
15 00:01:21,314 –> 00:01:26,931 and now you should see the values that come out are just the first 4 odd numbers.
16 00:01:28,057 –> 00:01:30,822 Hence 1, 3, 5, 7.
17 00:01:31,800 –> 00:01:35,671 For example 7 we get when we apply this 3 here.
18 00:01:36,700 –> 00:01:43,610 So you see this construction is not so complicated, but it occurs very often. Therefore you should know what it means.
19 00:01:44,600 –> 00:01:51,047 As a fun fact i can tell you this set construction is also included in the programming language Python.
20 00:01:52,229 –> 00:01:59,077 So you can use the curly brackets { and type “2 times x + 1 for
21 00:02:00,386 –> 00:02:03,408 So you can see the “for” as the vertical line.
22 00:02:04,329 –> 00:02:11,900 and then we just type “x in the set {0, 1, 2, 3} curly brackets.
23 00:02:11,957 –> 00:02:12,786 curly brackets
24 00:02:13,643 –> 00:02:16,729 Ok now i hit enter and you see the result.
25 00:02:17,957 –> 00:02:22,461 So not so surprising we get the same odd numbers, we calculated before.
26 00:02:23,443 –> 00:02:27,540 Ok i hope that this was helpful and that i see you in the next video.
27 00:02:27,740 –> 00:02:28,514 Bye!
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Quiz Content
Q1: How do mathematicians denote the set that contains the elements $1$ and $2$?
A1: ${ 1, 2 }$
A2: $[ 1, 2 ]$
A3: ${ { 1 } , {2} }$
A4: $(1,2)$
Q2: How to write the set that contains all even natural numbers?
A1: ${ 2 x \mid x \in \mathbb{N} }$.
A2: ${ 2 x +1 \mid x \in \mathbb{N} }$.
A3: ${ x +1 \mid x \in \mathbb{N} }$.
A4: ${ 2 x +1 \mid x \in \mathbb{Z} }$.
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Last update: 2024-11
