Composition operator - Revision history

en>Brirush: adding section headings

2014-11-23T13:03:45Z

adding section headings

← Older revision		Revision as of 15:03, 23 November 2014
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	~~{{About\|relational algebra\|other uses of the term "projection"\|Projection (disambiguation)}}~~		Greetings. The writer's title is Phebe and she feels comfortable when people use the full title. To collect badges is what her family members and her enjoy. North Dakota is where me and my husband reside. Managing people is his occupation.<br><br>Here is my blog post ... [http://www.youporntime.com/blog/12800 http://www.youporntime.com]
	~~{{Expert-subject\|Mathematics\|date=February 2009}}~~
	In [[relational algebra]], a '''projection''' is a [[unary operation]] written as <math>\Pi_{a_1, ...,a_n}( R )</math> where <math>a_1,...,a_n</math> is a set of attribute names. The result of such projection is defined as the [[Set (mathematics)\|set]] obtained when the components of the [[tuple]] <math>R</math> are restricted to the set <math>\{a_1,...,a_n\}</math> – it ''discards'' (or ''excludes'') the other attributes.<ref>http://www.cs.rochester.edu/~nelson/courses/csc_173/relations/algebra.html</ref>

	In practical terms, it can be roughly thought of as picking a sub-set of all available columns. For example, if the attributes are (name, age), then projection of the relation {(Alice, 5), (Bob, 8)} onto attribute list (age) yields {5,8} – we have discarded the names, and only know what ages are present.


	~~In addition, projection can be used to modify an attribute's value: if relation R has attributes a, b, and c, and b is a number, then~~
	~~<math>\Pi_{a,\ b * 0.5,\ c}( R )</math>~~
	~~will return a relation nearly the same as R, but with all values for 'b' shrunk by half. <ref>http://www.csee.umbc.edu/~pmundur/courses/CMSC661-02/rel-alg.pdf ''See Problem 3.8~~.~~B on page 3''</ref>~~


	~~==Related concepts==~~
	The closely related concept in [[set theory]] (see: [[projection (set theory)]]) differs from that of [[relational algebra]] in that, in set theory, one projects onto ordered components, not onto attributes. For instance, projecting <math>(3,7)</math> onto the second component yields 7.

	~~Projection is relational algebra~~'s counterpart of [[existential quantification]] in [[first-order logic\|predicate logic]]. The attributes ''not'' included correspond to existentially quantified variables in the predicate whose [[extension (predicate logic)\|extension]] the operand relation represents. The example below illustrates this point.

	~~Because of the correspondence with existential quantification, some authorities prefer to define projection in terms of the excluded attributes. In a computer language it~~ is ~~of course possible to provide notations for both,~~ and ~~that was done in [[ISBL]] and several languages that have taken their cue from ISBL.~~

	~~A nearly identical concept occurs in~~ the ~~category of [[monoid]]s, called a [[string projection]], which consists of removing all of the letters in the [[string (computer science)\|string]] that do not belong to a given [[alphabet]]~~.

	~~==Example==~~
	For an example, consider the relations depicted in the following two tables which are the relation <math>Person</math> and its projection on (some say "over") the attributes <math>Age</math> and <math>Weight</math>:

	~~{\| border="0" cellpadding="0" cellspacing="20" align="center"~~
	\|-
	~~! <math>Person</math>~~
	~~! <math>\Pi_{Age,Weight}(Person)</math>~~
	~~\|- style="vertical-align: top"~~
	\|
	~~{\| align="center" border="0" cellpadding="10" cellspacing="0" style="border-collapse: collapse; border: 1px solid black"~~
	~~\|- style="background-color: silver; text-align: left"~~
	~~! style="border: 1px solid black" width="34%" \| Name~~
	~~! style="border: 1px solid black" width="33%" \| Age~~
	~~! style="border: 1px solid black" width="33%" \| Weight~~
	\|-
	~~\| style="border: 1px solid black" \| Harry~~
	~~\| style="border: 1px solid black" \| 34~~
	~~\| style="border: 1px solid black" \| 180~~
	\|-
	~~\| style="border: 1px solid black" \| Sally~~
	~~\| style="border: 1px solid black" \| 28~~
	~~\| style="border: 1px solid black" \| 164~~
	\|-
	~~\| style="border: 1px solid black" \| George~~
	~~\| style="border: 1px solid black" \| 29~~
	~~\| style="border: 1px solid black" \| 170~~
	\|-
	~~\| style="border: 1px solid black" \| Helena~~
	~~\| style="border: 1px solid black" \| 54~~
	~~\| style="border: 1px solid black" \| 154~~
	\|-
	~~\| style="border: 1px solid black" \| Peter~~
	~~\| style="border: 1px solid black" \| 34~~
	~~\| style="border: 1px solid black" \| 180~~
	\|}
	\|
	~~{\| align="center" border="0" cellpadding="10" cellspacing="0" style="border-collapse: collapse; border: 1px solid black"~~
	~~\|- style="background-color: silver; text-align: left"~~
	~~! style="border: 1px solid black" width="50%" \| Age~~
	~~! style="border: 1px solid black" width="50%" \| Weight~~
	\|-
	~~\| style="border: 1px solid black" \| 34~~
	~~\| style="border: 1px solid black" \| 180~~
	\|-
	~~\| style="border: 1px solid black" \| 28~~
	~~\| style="border: 1px solid black" \| 164~~
	\|-
	~~\| style="border: 1px solid black" \| 29~~
	~~\| style="border: 1px solid black" \| 170~~
	\|-
	~~\| style="border: 1px solid black" \| 54~~
	~~\| style="border: 1px solid black" \| 154~~
	\|-
	\|}
	\|}

	~~Suppose the predicate of Person~~ is ~~"''Name'' is ''age'' years old~~ and ~~weighs ''weight''~~.~~" Then the given projection represents the predicate, "There exists ''Name'' such that ''Name''~~ is ~~''age'' years old~~ and ~~weighs ''weight''~~."

	~~Note that Harry and Peter have the same age and weight, but since the result~~ is ~~a relation, and therefore a set, this combination only appears once in the result.~~

	~~More formally the semantics of projection are defined as follows:~~

	~~: <math>\Pi_{a_1, ...,a_n}( R ) = \{ \ t[a_1,~~.~~..,a_n] : \ t \in R \ \}~~<~~/math~~>

	~~where~~ <~~math~~>~~t[a_1,~~...~~,a_n]</math> is the~~ [~~[restriction (mathematics)\|restriction]] of the tuple <math>t<~~/~~math> to the set <math>\{a_1,...,a_n\}<~~/~~math> so that~~

	~~: <math>t[a_1,~~..~~.,a_n] = \{ \ ( a', v ) \ \| \ ( a', v ) \in t, \ a' \in a_1,...,a_n \ \}<~~/~~math>~~

	~~The result of a projection <math>\Pi_{a_1, ...,a_n}( R )<~~/~~math> is defined only if <math>\{a_1,...,a_n\}<~~/~~math> is a [[subset]] of the [[:wikt:heading\|header]] of <math>R<~~/~~math>.~~

	It is interesting to note that projection over no attributes at all is possible, yielding a relation of degree zero. In this case the cardinality of the result is zero if the operand is empty, otherwise one. ~~The two relations of degree zero are the only ones that cannot be depicted as tables~~.

	~~==See also==~~
	* [[Projection (set theory)]]
	* [[Extended projection]]

	~~==References==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Projection (Relational Algebra)}}~~
	~~[[Category:Relational algebra]]~~

	~~[[ru:Алгебра Кодда#Проекция]~~]

en>Mark viking: Added wl

2014-01-11T04:10:48Z

Added wl

← Older revision		Revision as of 06:10, 11 January 2014
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	~~Greetings! I am Marvella and I really feel comfortable~~ when ~~individuals use~~ the ~~complete title~~. ~~South Dakota~~ is ~~where I~~'~~ve always been living~~. ~~One~~ of the ~~issues she enjoys most~~ is to ~~study comics and she'll be beginning something else alongside~~ with it. ~~Hiring is my occupation~~.<br><br>~~Feel free to visit my website~~ - [~~http~~://~~nemoonehfilm~~.ir/~~index~~.~~php?do~~=/~~profile-6876~~/~~info~~/ over the ~~counter std test~~]		{{About\|relational algebra\|other uses of the term "projection"\|Projection (disambiguation)}}
			{{Expert-subject\|Mathematics\|date=February 2009}}
			In [[relational algebra]], a '''projection''' is a [[unary operation]] written as <math>\Pi_{a_1, ...,a_n}( R )</math> where <math>a_1,...,a_n</math> is a set of attribute names. The result of such projection is defined as the [[Set (mathematics)\|set]] obtained when the components of the [[tuple]] <math>R</math> are restricted to the set <math>\{a_1,...,a_n\}</math> – it ''discards'' (or ''excludes'') the other attributes.<ref>http://www.cs.rochester.edu/~nelson/courses/csc_173/relations/algebra.html</ref>

			In practical terms, it can be roughly thought of as picking a sub-set of all available columns. For example, if the attributes are (name, age), then projection of the relation {(Alice, 5), (Bob, 8)} onto attribute list (age) yields {5,8} – we have discarded the names, and only know what ages are present.


			In addition, projection can be used to modify an attribute's value: if relation R has attributes a, b, and c, and b is a number, then
			<math>\Pi_{a,\ b * 0.5,\ c}( R )</math>
			will return a relation nearly the same as R, but with all values for 'b' shrunk by half. <ref>http://www.csee.umbc.edu/~pmundur/courses/CMSC661-02/rel-alg.pdf ''See Problem 3.8.B on page 3''</ref>


			==Related concepts==
			The closely related concept in [[set theory]] (see: [[projection (set theory)]]) differs from that of [[relational algebra]] in that, in set theory, one projects onto ordered components, not onto attributes. For instance, projecting <math>(3,7)</math> onto the second component yields 7.

			Projection is relational algebra's counterpart of [[existential quantification]] in [[first-order logic\|predicate logic]]. The attributes ''not'' included correspond to existentially quantified variables in the predicate whose [[extension (predicate logic)\|extension]] the operand relation represents. The example below illustrates this point.

			Because of the correspondence with existential quantification, some authorities prefer to define projection in terms of the excluded attributes. In a computer language it is of course possible to provide notations for both, and that was done in [[ISBL]] and several languages that have taken their cue from ISBL.

			A nearly identical concept occurs in the category of [[monoid]]s, called a [[string projection]], which consists of removing all of the letters in the [[string (computer science)\|string]] that do not belong to a given [[alphabet]].

			==Example==
			For an example, consider the relations depicted in the following two tables which are the relation <math>Person</math> and its projection on (some say "over") the attributes <math>Age</math> and <math>Weight</math>:

			{\| border="0" cellpadding="0" cellspacing="20" align="center"
			\|-
			! <math>Person</math>
			! <math>\Pi_{Age,Weight}(Person)</math>
			\|- style="vertical-align: top"
			\|
			{\| align="center" border="0" cellpadding="10" cellspacing="0" style="border-collapse: collapse; border: 1px solid black"
			\|- style="background-color: silver; text-align: left"
			! style="border: 1px solid black" width="34%" \| Name
			! style="border: 1px solid black" width="33%" \| Age
			! style="border: 1px solid black" width="33%" \| Weight
			\|-
			\| style="border: 1px solid black" \| Harry
			\| style="border: 1px solid black" \| 34
			\| style="border: 1px solid black" \| 180
			\|-
			\| style="border: 1px solid black" \| Sally
			\| style="border: 1px solid black" \| 28
			\| style="border: 1px solid black" \| 164
			\|-
			\| style="border: 1px solid black" \| George
			\| style="border: 1px solid black" \| 29
			\| style="border: 1px solid black" \| 170
			\|-
			\| style="border: 1px solid black" \| Helena
			\| style="border: 1px solid black" \| 54
			\| style="border: 1px solid black" \| 154
			\|-
			\| style="border: 1px solid black" \| Peter
			\| style="border: 1px solid black" \| 34
			\| style="border: 1px solid black" \| 180
			\|}
			\|
			{\| align="center" border="0" cellpadding="10" cellspacing="0" style="border-collapse: collapse; border: 1px solid black"
			\|- style="background-color: silver; text-align: left"
			! style="border: 1px solid black" width="50%" \| Age
			! style="border: 1px solid black" width="50%" \| Weight
			\|-
			\| style="border: 1px solid black" \| 34
			\| style="border: 1px solid black" \| 180
			\|-
			\| style="border: 1px solid black" \| 28
			\| style="border: 1px solid black" \| 164
			\|-
			\| style="border: 1px solid black" \| 29
			\| style="border: 1px solid black" \| 170
			\|-
			\| style="border: 1px solid black" \| 54
			\| style="border: 1px solid black" \| 154
			\|-
			\|}
			\|}

			Suppose the predicate of Person is "''Name'' is ''age'' years old and weighs ''weight''." Then the given projection represents the predicate, "There exists ''Name'' such that ''Name'' is ''age'' years old and weighs ''weight''."

			Note that Harry and Peter have the same age and weight, but since the result is a relation, and therefore a set, this combination only appears once in the result.

			More formally the semantics of projection are defined as follows:

			: <math>\Pi_{a_1, ...,a_n}( R ) = \{ \ t[a_1,...,a_n] : \ t \in R \ \}</math>

			where <math>t[a_1,...,a_n]</math> is the [[restriction (mathematics)\|restriction]] of the tuple <math>t</math> to the set <math>\{a_1,...,a_n\}</math> so that

			: <math>t[a_1,...,a_n] = \{ \ ( a', v ) \ \| \ ( a', v ) \in t, \ a' \in a_1,...,a_n \ \}</math>

			The result of a projection <math>\Pi_{a_1, ...,a_n}( R )</math> is defined only if <math>\{a_1,...,a_n\}</math> is a [[subset]] of the [[:wikt:heading\|header]] of <math>R</math>.

			It is interesting to note that projection over no attributes at all is possible, yielding a relation of degree zero. In this case the cardinality of the result is zero if the operand is empty, otherwise one. The two relations of degree zero are the only ones that cannot be depicted as tables.

			==See also==
			* [[Projection (set theory)]]
			* [[Extended projection]]

			==References==
			{{reflist}}

			{{DEFAULTSORT:Projection (Relational Algebra)}}
			[[Category:Relational algebra]]

			[[ru:Алгебра Кодда#Проекция]]

en>Linas: /* See also */ Composition ring

2012-08-26T20:02:40Z