Mathematics/Discrete Math/Sets - Revision history

Brodriguez: Brodriguez moved page Discrete Math/Sets to Mathematics/Discrete Math/Sets

2020-10-25T17:23:25Z

Brodriguez moved page Discrete Math/Sets to Mathematics/Discrete Math/Sets

← Older revision

Revision as of 17:23, 25 October 2020

Brodriguez: Brodriguez moved page Discrete Math - Sets to Discrete Math/Sets: Clean url with subpages

2020-05-15T08:55:00Z

Brodriguez moved page Discrete Math - Sets to Discrete Math/Sets: Clean url with subpages

← Older revision

Revision as of 08:55, 15 May 2020

Brodriguez: Add disjoint definition

2019-09-10T16:37:29Z

Add disjoint definition

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	* '''Intersection''' of A and B ('''A ∩ B''') - The set of all distinct elements that are contained in both A and B.		* '''Intersection''' of A and B ('''A ∩ B''') - The set of all distinct elements that are contained in both A and B.
	** Ex: <code>S={2, 4}</code>		** Ex: <code>S={2, 4}</code>
			* '''Disjoint''' - Two sets whose intersection is the empty set.
	* '''Difference''' ('''A - B''') - The given set, minus all elements in the second set.		* '''Difference''' ('''A - B''') - The given set, minus all elements in the second set.
	** Ex: <code>A - B={3, 5}</code>		** Ex: <code>A - B={3, 5}</code>

Brodriguez: Correct minor errors

2019-09-08T03:45:05Z

Correct minor errors

Brodriguez: Add definition of element

2019-09-08T00:57:15Z

Add definition of element

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	== Terminology ==		== Terminology ==
			* '''Element''' - A single item to put within a set. May be a number, an object, or in rarer circumstances, another set.
	* '''Set''' - A grouping of various elements.		* '''Set''' - A grouping of various elements.
	* '''Subset''' - A grouping of such that all elements of the set are also contained within an equal-sized or larger set. May include two equivalent sets.		* '''Subset''' - A grouping of such that all elements of the set are also contained within an equal-sized or larger set. May include two equivalent sets.

Brodriguez: Add set notation section

2019-09-08T00:55:09Z

Add set notation section

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	*** For example, a set of Cardinality 3 will have 2<sup>3</sup> elements.		*** For example, a set of Cardinality 3 will have 2<sup>3</sup> elements.
	** Ex: <code>S={∅, {2}, {3}, {4}, {5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}, {2, 3, 4, 5}}</code>		** Ex: <code>S={∅, {2}, {3}, {4}, {5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}, {2, 3, 4, 5}}</code>

			== Set-Builder Notation ==
			Sometimes referred to as "Set Notation". Set-Builder Notation is a standardized way to describe a given set.
			<br>
			The following are explained using the arbitrary sets <code>A</code> and <code>B</code> plus the arbitrary elements <code>x</code> and <code>x</code>.
			* '''x ∈ A''' - Means that element <code>x</code> is in set <code>A</code>.
			* '''x ∉ A''' - Means that element <code>x</code> is not in set <code>A</code>.
			* '''A ⊆ B''' - Means that <code>A</code> is a Subset of <code>B</code>. In other words, all elements in set <code>A</code> are contained within set <code>B</code>. Set <code>B</code> may or may not have additional elements past what what <code>A</code> contains, potentially resulting in <code>A = B</code>.
			* '''A ⊆ B''' - Means that <code>A</code> is a Proper Subset of <code>B</code>. In other words, all elements in set <code>A</code> are contained within set <code>B</code>. Set <code>B</code> must also have additional elements past what what <code>A</code> contains, resulting in <code>A ≠ B</code>.
			* '''\|''' or ''':''' - "Such that...".
			** Ex: <code>{ q \| q > 0 }</code> - Translates to "The set of all possible values for q, such that q is greater than zero."
			** Ex: <code>{ x ∈ R \| x ≥ 5 }</code> - Translates to "The set of all Real Numbers x, such that x is greater than or equal to 5."
			* '''∀''' - "For all...".
			** Ex: <code>∀x ∈ Z<sup>+</sup>, x+1 < x+2</code> - Translates to "x+1 is less than x+2 for all x, as long as x is a positive integer.".
			* '''∃''' - "There exists at least one...".
			** Ex: <code>∃x \| P(x)</code> - Translates to "There exists at least one x, such that the function P(x) is true.
			* * '''∃!''' - "There exists exactly one...".
			* '''∴''' - "Therefore...".

Brodriguez: Improve formatting

2019-09-07T22:48:26Z

Improve formatting

← Older revision		Revision as of 22:48, 7 September 2019
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	== Terminology ==		== Terminology ==
	* Set - A grouping of various elements.		* '''Set''' - A grouping of various elements.
	* Subset - A grouping of such that all elements of the set are also contained within an equal-sized or larger set. May include two equivalent sets.		* '''Subset''' - A grouping of such that all elements of the set are also contained within an equal-sized or larger set. May include two equivalent sets.
	* Proper Subset - A subset in which the two sets are not equivalent.		* '''Proper Subset''' - A subset in which the two sets are not equivalent.
	* Empty Set/Zero Set ('''∅''') - A set containing exactly 0 elements. Aka a null set.		* '''Empty Set/Zero Set''' ('''∅''') - A set containing exactly 0 elements. Aka a null set.
	* Closed Set - A set in which it's own boundary is contained within the set.		* '''Closed Set''' - A set in which it's own boundary is contained within the set.
	* Open Set - A set where the boundary is excluded from the set, but all values between the boundary are included.		* '''Open Set''' - A set where the boundary is excluded from the set, but all values between the boundary are included.
	* Cardinality - The number of distinct elements within a set. Ex: The set ~~'''~~{-1, 0, 1}~~'''~~ has a Cardinality of 3.		* '''Cardinality''' - The number of distinct elements within a set.
			** Ex: The set <code>{-1, 0, 1}</code> has a Cardinality of 3.
	<br>		<br>
	Common Set Types:		Common Set Types:
	* Universal Set ('''U''') - An arbitrary set, which changes based on context. However, it is assumed to always contain every possible element currently being considered for the given context.		* '''Universal Set''' ('''U''') - An arbitrary set, which changes based on context. However, it is assumed to always contain every possible element currently being considered for the given context.
	* Integers ('''Z''') - The set of all integers. Ex: ~~'''~~{..., -3, -2, -1, 0, 1, 2, 3, ...}'''.		* '''Integers''' ('''Z''') - The set of all integers.
	* Positive Integers ('''Z<sup>+</sup>''') - The set of all positive integers. Ex: ~~'''~~{1, 2, 3, ...}'''.		** Ex: <code>{..., -3, -2, -1, 0, 1, 2, 3, ...}</code>.
	* Natural Numbers ('''N''') - The set of '''Z<sup>+</sup>''', but including zero. Ex: ~~'''~~{0, 1, 2, 3, ...}'''		* '''Positive Integers''' ('''Z<sup>+</sup>''') - The set of all positive integers.
	* Positive Real Numbers ('''R+''') - The set of non-imaginary numbers, greater than zero. Includes fractions, decimals, etc.		** Ex: <code>{1, 2, 3, ...}</code>.
	* Real Numbers ('''R''') - The set of '''R+''', but including zero and negative numbers.		* '''Natural Numbers''' ('''N''') - The set of '''Z<sup>+</sup>''', but including zero.
	* Complex Numbers ('''C''') - The set of all numbers, including imaginary ones.		** Ex: <code>{0, 1, 2, 3, ...}</code>.
			* '''Positive Real Numbers''' ('''R+''') - The set of non-imaginary numbers, greater than zero. Includes fractions, decimals, etc.
			* '''Real Numbers''' ('''R''') - The set of '''R+''', but including zero and negative numbers.
			* '''Complex Numbers''' ('''C''') - The set of all numbers, including imaginary ones.

	== Basic Set Operations ==		== Basic Set Operations ==
	The following are basic operations are explained with sets ~~'''~~A={2, 3, 4, 5}~~'''~~ and ~~'''~~B={0, 2, 4, 6}~~'''~~ onto set ~~'''~~S~~'''~~. But the operations could be applied to any arbitrary two sets.		The following are basic operations are explained with sets <code>A={2, 3, 4, 5}</code> and <code>B={0, 2, 4, 6}</code> onto set <code>S</code>. But the operations could be applied to any arbitrary two sets.
	* Union of A and B ('''A ∪ B''') - The set of all distinct elements contained in either A or B.		* '''Union''' of A and B ('''A ∪ B''') - The set of all distinct elements contained in either A or B.
	** Ex: ~~'''~~S={0, 2, 3, 4 ,5, 6}'''.		** Ex: <code>S={0, 2, 3, 4 ,5, 6}</code>.
	* Intersection of A and B ('''A ∩ B''') - The set of all distinct elements that are contained in both A and B.		* '''Intersection''' of A and B ('''A ∩ B''') - The set of all distinct elements that are contained in both A and B.
	** Ex: ~~'''~~S={2, 4}'''.		** Ex: <code>S={2, 4}</code>
	* Difference - The given set, minus all elements in the second set.		* '''Difference''' ('''A/B''') - The given set, minus all elements in the second set.
	** Ex: ~~'''~~A/B={3, 5}~~'''~~		** Ex: <code>A/B={3, 5}</code>
	** Ex: ~~'''~~B/A={0, 6}'''		** Ex: <code>B/A={0, 6}</code>
	* Symmetric Difference of A and B ('''A △ B''' or '''A ⊖ B''') - The set of all elements that are a member of exactly one set A or B.		* '''Symmetric Difference''' of A and B ('''A △ B''' or '''A ⊖ B''') - The set of all elements that are a member of exactly one set A or B.
	** Ex: ~~'''~~S={0, 3, 5, 6}'''		** Ex: <code>S={0, 3, 5, 6}</code>.
	* Cartesian Product of A and B (''' A x B''') - The set containing all ordered pairs ~~'''~~(a, b)~~'''~~ for each ~~'''~~a~~'''~~ in set A and each ~~'''~~b~~'''~~ in set B.		* '''Cartesian Product''' of A and B (''' A x B''') - The set containing all ordered pairs <code>(a, b)</code> for each <code>a</code> in set <code>A</code> and each <code>b</code> in set <code>B</code>.
	** Ex: ~~'''~~S={(2, 0), (2, 2), (2, 4), (2, 6), (3, 0), (3, 2), (3, 4), (3, 6), (4, 0), (4, 2), (4, 4), (4, 6), (5, 0), (5, 2), (5, 4), (5, 6)}'''		** Ex: <code>S={(2, 0), (2, 2), (2, 4), (2, 6), (3, 0), (3, 2), (3, 4), (3, 6), (4, 0), (4, 2), (4, 4), (4, 6), (5, 0), (5, 2), (5, 4), (5, 6)}</code>
	* Power Set of A ('''A''') - The set containing all possible subsets using elements from the original set A. Includes the empty set.		* '''Power Set''' of A ('''A''') - The set containing all possible subsets using elements from the original set A. Includes the empty set.
	** Note that the number of elements is always equal to 2<sup>x</sup>, where x is equal to the cardinality of the given set. This is why it's called the power set; The number of elements corresponds to powers of 2. Thus, counting the elements is a good way to check that it's correct.		** Note that the number of elements is always equal to 2<sup>x</sup>, where x is equal to the cardinality of the given set. This is why it's called the power set; The number of elements corresponds to powers of 2. Thus, counting the elements is a good way to check that it's correct.
	*** For example, a set of Cardinality 3 will have 2<sup>3</sup> elements.		*** For example, a set of Cardinality 3 will have 2<sup>3</sup> elements.
	** Ex: ~~'''~~S={∅, {2}, {3}, {4}, {5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}, {2, 3, 4, 5}}~~'''~~		** Ex: <code>S={∅, {2}, {3}, {4}, {5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}, {2, 3, 4, 5}}</code>

Brodriguez: Add basic set operations

2019-09-07T22:21:02Z

Add basic set operations

← Older revision		Revision as of 22:21, 7 September 2019
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	* Real Numbers ('''R''') - The set of '''R+''', but including zero and negative numbers.		* Real Numbers ('''R''') - The set of '''R+''', but including zero and negative numbers.
	* Complex Numbers ('''C''') - The set of all numbers, including imaginary ones.		* Complex Numbers ('''C''') - The set of all numbers, including imaginary ones.

			== Basic Set Operations ==
			The following are basic operations are explained with sets '''A={2, 3, 4, 5}''' and '''B={0, 2, 4, 6}''' onto set '''S'''. But the operations could be applied to any arbitrary two sets.
			* Union of A and B ('''A ∪ B''') - The set of all distinct elements contained in either A or B.
			** Ex: '''S={0, 2, 3, 4 ,5, 6}'''.
			* Intersection of A and B ('''A ∩ B''') - The set of all distinct elements that are contained in both A and B.
			** Ex: '''S={2, 4}'''.
			* Difference - The given set, minus all elements in the second set.
			** Ex: '''A/B={3, 5}'''
			** Ex: '''B/A={0, 6}'''
			* Symmetric Difference of A and B ('''A △ B''' or '''A ⊖ B''') - The set of all elements that are a member of exactly one set A or B.
			** Ex: '''S={0, 3, 5, 6}'''
			* Cartesian Product of A and B (''' A x B''') - The set containing all ordered pairs '''(a, b)''' for each '''a''' in set A and each '''b''' in set B.
			** Ex: '''S={(2, 0), (2, 2), (2, 4), (2, 6), (3, 0), (3, 2), (3, 4), (3, 6), (4, 0), (4, 2), (4, 4), (4, 6), (5, 0), (5, 2), (5, 4), (5, 6)}'''
			* Power Set of A ('''A''') - The set containing all possible subsets using elements from the original set A. Includes the empty set.
			** Note that the number of elements is always equal to 2<sup>x</sup>, where x is equal to the cardinality of the given set. This is why it's called the power set; The number of elements corresponds to powers of 2. Thus, counting the elements is a good way to check that it's correct.
			*** For example, a set of Cardinality 3 will have 2<sup>3</sup> elements.
			** Ex: '''S={∅, {2}, {3}, {4}, {5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}, {2, 3, 4, 5}}'''

Brodriguez: Add common set types

2019-09-07T20:22:03Z

Add common set types

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	* Open Set - A set where the boundary is excluded from the set, but all values between the boundary are included.		* Open Set - A set where the boundary is excluded from the set, but all values between the boundary are included.
	* Cardinality - The number of distinct elements within a set. Ex: The set '''{-1, 0, 1}''' has a Cardinality of 3.		* Cardinality - The number of distinct elements within a set. Ex: The set '''{-1, 0, 1}''' has a Cardinality of 3.
			<br>
			Common Set Types:
			* Universal Set ('''U''') - An arbitrary set, which changes based on context. However, it is assumed to always contain every possible element currently being considered for the given context.
			* Integers ('''Z''') - The set of all integers. Ex: '''{..., -3, -2, -1, 0, 1, 2, 3, ...}'''.
			* Positive Integers ('''Z<sup>+</sup>''') - The set of all positive integers. Ex: '''{1, 2, 3, ...}'''.
			* Natural Numbers ('''N''') - The set of '''Z<sup>+</sup>''', but including zero. Ex: '''{0, 1, 2, 3, ...}'''
			* Positive Real Numbers ('''R+''') - The set of non-imaginary numbers, greater than zero. Includes fractions, decimals, etc.
			* Real Numbers ('''R''') - The set of '''R+''', but including zero and negative numbers.
			* Complex Numbers ('''C''') - The set of all numbers, including imaginary ones.

Brodriguez: Create page

2019-09-07T20:21:41Z

Create page

New page

Sets, aka groups of elements and logic regarding such.

== Terminology ==
* Set - A grouping of various elements.
* Subset - A grouping of such that all elements of the set are also contained within an equal-sized or larger set. May include two equivalent sets.
* Proper Subset - A subset in which the two sets are not equivalent.
* Empty Set/Zero Set ('''∅''') - A set containing exactly 0 elements. Aka a null set.
* Closed Set - A set in which it's own boundary is contained within the set.
* Open Set - A set where the boundary is excluded from the set, but all values between the boundary are included.
* Cardinality - The number of distinct elements within a set. Ex: The set '''{-1, 0, 1}''' has a Cardinality of 3.

← Older revision		Revision as of 03:45, 8 September 2019
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	* '''Intersection''' of A and B ('''A ∩ B''') - The set of all distinct elements that are contained in both A and B.		* '''Intersection''' of A and B ('''A ∩ B''') - The set of all distinct elements that are contained in both A and B.
	** Ex: <code>S={2, 4}</code>		** Ex: <code>S={2, 4}</code>
	* '''Difference''' ('''A/B''') - The given set, minus all elements in the second set.		* '''Difference''' ('''A - B''') - The given set, minus all elements in the second set.
	** Ex: <code>A/B={3, 5}</code>		** Ex: <code>A - B={3, 5}</code>
	** Ex: <code>B/A={0, 6}</code>		** Ex: <code>B - A={0, 6}</code>
	* '''Symmetric Difference''' of A and B ('''A △ B''' or '''A ⊖ B''') - The set of all elements that are a member of exactly one set A or B.		* '''Symmetric Difference''' of A and B ('''A △ B''' or '''A ⊖ B''') - The set of all elements that are a member of exactly one set A or B.
	** Ex: <code>S={0, 3, 5, 6}</code>.		** Ex: <code>S={0, 3, 5, 6}</code>.
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	* '''x ∉ A''' - Means that element <code>x</code> is not in set <code>A</code>.		* '''x ∉ A''' - Means that element <code>x</code> is not in set <code>A</code>.
	* '''A ⊆ B''' - Means that <code>A</code> is a Subset of <code>B</code>. In other words, all elements in set <code>A</code> are contained within set <code>B</code>. Set <code>B</code> may or may not have additional elements past what what <code>A</code> contains, potentially resulting in <code>A = B</code>.		* '''A ⊆ B''' - Means that <code>A</code> is a Subset of <code>B</code>. In other words, all elements in set <code>A</code> are contained within set <code>B</code>. Set <code>B</code> may or may not have additional elements past what what <code>A</code> contains, potentially resulting in <code>A = B</code>.
	* '''A ⊆ B''' - Means that <code>A</code> is a Proper Subset of <code>B</code>. In other words, all elements in set <code>A</code> are contained within set <code>B</code>. Set <code>B</code> must also have additional elements past what what <code>A</code> contains, resulting in <code>A ≠ B</code>.		* '''A ⊂ B''' - Means that <code>A</code> is a Proper Subset of <code>B</code>. In other words, all elements in set <code>A</code> are contained within set <code>B</code>. Set <code>B</code> must also have additional elements past what what <code>A</code> contains, resulting in <code>A ≠ B</code>.
	* '''\|''' or ''':''' - "Such that...".		* '''\|''' or ''':''' - "Such that...".
	** Ex: <code>{ q \| q > 0 }</code> - Translates to "The set of all possible values for q, such that q is greater than zero."		** Ex: <code>{ q \| q > 0 }</code> - Translates to "The set of all possible values for q, such that q is greater than zero."