View source for Mathematics/Discrete Math/Sets

Sets, aka groups of elements and logic regarding such.

== 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.
* '''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 <code>{-1, 0, 1}</code> has a Cardinality of 3.
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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: <code>{..., -3, -2, -1, 0, 1, 2, 3, ...}</code>.
* '''Positive Integers''' ('''Z<sup>+</sup>''') - The set of all positive integers.
** Ex: <code>{1, 2, 3, ...}</code>.
* '''Natural Numbers''' ('''N''') - The set of '''Z<sup>+</sup>''', but including zero.
** 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 ==
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.
** 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.
** Ex: <code>S={2, 4}</code>
* '''Difference''' ('''A/B''') - The given set, minus all elements in the second set.
** Ex: <code>A/B={3, 5}</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.
** Ex: <code>S={0, 3, 5, 6}</code>.
* '''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: <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.
** 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: <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.
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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...".