Set notation - Revision history

Mscroggs at 12:30, 13 July 2021

2021-07-13T12:30:04Z

← Older revision		Revision as of 13:30, 13 July 2021
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	$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$		$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$

			==Bars==

			Bars can experience issues with ambiguity too, for example:

			$$\{a\in\mathbb{R}\mid\|a\|<2\}$$
			$$\{a\in\mathbb{Z}\mid\|a\|\vert12\}$$

Christian Lawson-Perfect at 09:43, 13 July 2021

2021-07-13T09:43:18Z

← Older revision		Revision as of 10:43, 13 July 2021
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	* All the elements of the set, separated by commas: $ \{a,b,c, \ldots\} $		* All the elements of the set, separated by commas: $ \{a,b,c, \ldots\} $
	* Set-builder notation: a name for a general element of the set, then a condition that all elements must satisfy, with either a vertical bar or a colon in between: $ \{ x \mid P(x) \} $ or $ \{ x : P(x) \} $.		* Set-builder notation: a name for a general element of the set, then a condition that all elements must satisfy, with either a vertical bar or a colon in between: $ \{ x \mid P(x) \} $ or $ \{ x : P(x) \} $.

			==What goes between braces?==

	Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?		Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?

	The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the $a_i$.		[[User:Hpecora1]]: The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the $a_i$.

			==Colons==

	Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg $f:\mathbb{R}\to\mathbb{R}$. This can cause ambiguities when defining a set containing functions, for example:		Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg $f:\mathbb{R}\to\mathbb{R}$. This can cause ambiguities when defining a set containing functions, for example:

	$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$		$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$

Hpecora1: Fixed TeX in paragraph I added earlier.

2021-07-12T16:47:27Z

Fixed TeX in paragraph I added earlier.

← Older revision		Revision as of 17:47, 12 July 2021
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	Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?		Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?

	The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the \a_i.		The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the $a_i$.

	Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg $f:\mathbb{R}\to\mathbb{R}$. This can cause ambiguities when defining a set containing functions, for example:		Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg $f:\mathbb{R}\to\mathbb{R}$. This can cause ambiguities when defining a set containing functions, for example:

	$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$		$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$

Hpecora1 at 16:30, 12 July 2021

2021-07-12T16:30:21Z

← Older revision		Revision as of 17:30, 12 July 2021
Line 8:		Line 8:

	Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?		Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?

			The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the \a_i.

	Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg $f:\mathbb{R}\to\mathbb{R}$. This can cause ambiguities when defining a set containing functions, for example:		Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg $f:\mathbb{R}\to\mathbb{R}$. This can cause ambiguities when defining a set containing functions, for example:

	$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$		$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$

Mscroggs at 07:48, 6 July 2021

2021-07-06T07:48:53Z

← Older revision		Revision as of 08:48, 6 July 2021
Line 8:		Line 8:

	Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?		Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?

			Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg $f:\mathbb{R}\to\mathbb{R}$. This can cause ambiguities when defining a set containing functions, for example:

			$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$

Christian Lawson-Perfect at 09:53, 5 July 2021

2021-07-05T09:53:13Z

← Older revision		Revision as of 10:53, 5 July 2021
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			[[Category:Local variations]]
			[[Category:Ambiguities]]

	When defining a set, something is normally put between curly braces. There are a few different conventions for what can go inside:		When defining a set, something is normally put between curly braces. There are a few different conventions for what can go inside:

Christian Lawson-Perfect: Created page with "When defining a set, something is normally put between curly braces. There are a few different conventions for what can go inside: * All the elements of the set, separated by..."

2021-07-05T09:52:38Z

Created page with "When defining a set, something is normally put between curly braces. There are a few different conventions for what can go inside: * All the elements of the set, separated by..."

New page

When defining a set, something is normally put between curly braces. There are a few different conventions for what can go inside:

* All the elements of the set, separated by commas: $ \{a,b,c, \ldots\} $
* Set-builder notation: a name for a general element of the set, then a condition that all elements must satisfy, with either a vertical bar or a colon in between: $ \{ x \mid P(x) \} $ or $ \{ x : P(x) \} $.

Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?

← Older revision		Revision as of 10:43, 13 July 2021
Line 6:		Line 6:
	* All the elements of the set, separated by commas: \( \{a,b,c, \ldots\} \)		* All the elements of the set, separated by commas: \( \{a,b,c, \ldots\} \)
	* Set-builder notation: a name for a general element of the set, then a condition that all elements must satisfy, with either a vertical bar or a colon in between: \( \{ x \mid P(x) \} \) or \( \{ x : P(x) \} \).		* Set-builder notation: a name for a general element of the set, then a condition that all elements must satisfy, with either a vertical bar or a colon in between: \( \{ x \mid P(x) \} \) or \( \{ x : P(x) \} \).

			==What goes between braces?==

	Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation \( \{ a_1, a_2, \ldots \} \), where the elements inside are a sequence, are you making an implicit claim that the elements are countable?		Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation \( \{ a_1, a_2, \ldots \} \), where the elements inside are a sequence, are you making an implicit claim that the elements are countable?

	The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the \(a_i\).		[[User:Hpecora1]]: The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the \(a_i\).

			==Colons==

	Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg \(f:\mathbb{R}\to\mathbb{R}\). This can cause ambiguities when defining a set containing functions, for example:		Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg \(f:\mathbb{R}\to\mathbb{R}\). This can cause ambiguities when defining a set containing functions, for example:

	$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$		$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$