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In mathematics, an injective function (also known as injection, or one-to-one function) is a function $f$ that maps distinct elements to distinct elements; that is, $f (x 1) = f (x 2)$ implies $x 1 = x 2$ . (Equivalently, $x 1 \neq x 2$ implies $f (x 1) Templat:\neq f (x 2)$ in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of at most one element of its domain.^[1] The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an injective homomorphism is also called a monomorphism. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism.^[2] This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details.

A function $f$ that is not injective is sometimes called many-to-one.^[1]

Definition

Let $f$ be a function whose domain is a set $X.$ The function $f$ is said to be injective provided that for all $a$ and $b$ in $X,$ if $f(a)=f(b),$ then $a=b$ ; that is, $f(a)=f(b)$ implies $a=b.$ Equivalently, if $a\neq b,$ then $f(a)\neq f(b)$ in the contrapositive statement.

Symbolically, $\forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,$ which is logically equivalent to the contrapositive,^[3] $\forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).$

^ ^a ^b "Injective, Surjective and Bijective". www.mathsisfun.com. Diakses tanggal 2019-12-07.
^ "Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project". stacks.math.columbia.edu. Diakses tanggal 2019-12-07.
^ Farlow, S. J. "Injections, Surjections, and Bijections" (PDF). math.umaine.edu. Diakses tanggal 2019-12-06.

[:0-1] "Injective, Surjective and Bijective". www.mathsisfun.com. Diakses tanggal 2019-12-07.

[2] "Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project". stacks.math.columbia.edu. Diakses tanggal 2019-12-07.

[3] Farlow, S. J. "Injections, Surjections, and Bijections" (PDF). math.umaine.edu. Diakses tanggal 2019-12-06.

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[2]

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@@ Baris 1: / Baris 1: @@
+{{functions}}In [[mathematics]], an '''injective function''' (also known as '''injection''', or '''one-to-one function''') is a [[function (mathematics)|function]] {{math|''f''}} that maps [[Distinct (mathematics)|distinct]] elements to distinct elements; that is, {{math|''f''(''x''<sub>1</sub>) {{=}} ''f''(''x''<sub>2</sub>)}} implies {{math|1=''x''<sub>1</sub> = ''x''<sub>2</sub>}}. (Equivalently, {{math|1=''x''<sub>1</sub> ≠ ''x''<sub>2</sub>}} implies {{math|''f''(''x''<sub>1</sub>) {{≠}} ''f''(''x''<sub>2</sub>)}} in the equivalent [[Contraposition|contrapositive]] statement.) In other words, every element of the function's [[codomain]] is the [[Image (mathematics)|image]] of {{em|at most}} one element of its [[Domain of a function|domain]].<ref name=":0">{{Cite web|url=https://www.mathsisfun.com/sets/injective-surjective-bijective.html|title=Injective, Surjective and Bijective|website=www.mathsisfun.com|access-date=2019-12-07}}</ref> The term {{em|one-to-one function}} must not be confused with {{em|one-to-one correspondence}} that refers to [[bijective function]]s, which are functions such that each element in the codomain is an image of exactly one element in the domain.
-{{short description|English clergyman, mathematician, geometer and astronomer}}
-'''Edmund Gunter''' adalah seorang keturunan Wales yang bekerja sebagai seorang pastor, matematikawan, ahli geometri dan [[astronom]] di Inggris.<ref>Guy O. Stenstrom (1967), "Surveying Ready Reference Manual", McGraw–Hill. p. 7</ref>  Ia dikenal utamanya karena jasanya dalam matematika, di antaranya penemuannya seperti [[rantai Gunter]], [[#Gunter's quadrant|kuadran Gunter]], dan [[#Gunter's scale|skala Gunter]]. Pada tahun 1620, ia berhasil mengembangkan [[perangkat analog]] pertama<ref>Trevor Homer (2012). "The Book of Origins: The first of everything – from art to zoos". Hachette UK</ref> yang ia kembangkan untuk menghitung tangen logaritmik.<ref>Eli Maor (2013). "Trigonometric Delights", Princeton University Press.</ref>
+A [[homomorphism]] between [[algebraic structure]]s is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for [[vector space]]s, an {{em|injective homomorphism}} is also called a {{em|[[monomorphism]]}}. However, in the more general context of [[category theory]], the definition of a monomorphism differs from that of an injective homomorphism.<ref>{{Cite web|url=https://stacks.math.columbia.edu/tag/00V5|title=Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project|website=stacks.math.columbia.edu|access-date=2019-12-07}}</ref> This is thus a theorem that they are equivalent for algebraic structures; see {{slink|Homomorphism|Monomorphism}} for more details.
-Ia didampingi dalam belajar matematika oleh Reverend [[Henry Briggs (mathematician)|Henry Briggs]] dan kemudian menjadi [[profesor astronomi di Gresham]], dari tahun 1619 hingga kematiannya.<ref>William E. Burns (2001), ''The Scientific Revolution: An Encyclopedia'', ABC-CLIO, p. 125</ref>
+A function <math>f</math> that is not injective is sometimes called many-to-one.<ref name=":0" />
-==Biography==
-Gunter was born in Hertfordshire in 1581. He was educated at [[Westminster School]], and in 1599 he matriculated at [[Christ Church, Oxford]]. He took orders, became a preacher in 1614, and in 1615 proceeded to the degree of bachelor in [[Divinity (academic discipline)|divinity]].<ref name="EB1911">{{EB1911|inline=y|wstitle=Gunter, Edmund|volume=12|pages=729–730}}</ref> He became rector of [[St George the Martyr Southwark|St. George's Church]] in Southwark.<ref>Christopher Baker (2002). "Absolutism and the Scientific Revolution, 1600–1720". Greenwood Publishing Group</ref>
+== Definition ==
-Mathematics, particularly the relationship between mathematics and the real world, was the one overriding interest throughout his life. In 1619, [[Henry Savile (Bible translator)|Sir Henry Savile]] put up money to fund Oxford University's first two science faculties, the chairs of astronomy and geometry. Gunter applied to become professor of geometry but Savile was famous for distrusting clever people, and Gunter's behaviour annoyed him intensely. As was his habit, Gunter arrived with his [[sector (instrument)|sector]] and [[Quadrant (instrument)|quadrant]], and began demonstrating how they could be used to calculate the position of stars or the distance of churches, until Savile could stand it no longer. "Doe you call this reading of Geometric?" he burst out. "This is mere showing of tricks, man!" and, according to a contemporary account, "dismissed him with scorne."<ref name=who>[http://www.gresham.ac.uk/event.asp?PageId=4&EventId=385 "Who invented the calculus? – and other 17th century topics"] {{webarchive|url=https://web.archive.org/web/20070928035913/http://www.gresham.ac.uk/event.asp?PageId=4&EventId=385 |date=28 September 2007 }}, Professor Robin Wilson, lecture transcript, [[Gresham College]], 16 November 2005. Retrieved 7 November 2010.</ref><ref>Linklater, Andro, ''Measuring America'', Penguin Books, 2003, p. 14</ref>
+{{Further|topic=notation|Function (mathematics)#Notation}}
-He was shortly thereafter championed by the far wealthier [[John Egerton, 1st Earl of Bridgewater|Earl of Bridgewater]], who saw to it that on 6 March 1619 Gunter was appointed professor of [[astronomy]] in [[Gresham College]], London. This post he held till his death.<ref name="EB1911"/>
+Let <math>f</math> be a function whose domain is a set <math>X.</math> The function <math>f</math> is said to be '''injective''' provided that for all <math>a</math> and <math>b</math> in <math>X,</math> if <math>f(a) = f(b),</math> then <math>a = b</math>; that is, <math>f(a) = f(b)</math> implies <math>a=b.</math> Equivalently, if <math>a \neq b,</math> then <math>f(a) \neq f(b)</math> in the [[Contraposition|contrapositive]] statement.
-With Gunter's name are associated several useful inventions, descriptions of which are given in his treatises on the sector, [[cross-staff]], [[Backstaff#Cross bow quadrant|bow]], [[quadrant (instrument)|quadrant]] and other instruments. He contrived his sector about the year 1606, and wrote a description of it in Latin, but it was more than sixteen years afterwards before he allowed the book to appear in English. In 1620 he published his ''Canon triangulorum''.<ref name="EB1911"/>{{efn|The site http://locomat.loria.fr contains a complete reconstruction of Gunter's book and table.}}
+Symbolically,<math display="block">\forall a,b \in X, \;\; f(a)=f(b) \Rightarrow a=b,</math>
-In 1624 Gunter published a collection of his mathematical works. It was entitled ''The description and use of sector, the cross-staffe, and other instruments for such as are studious of mathematical practise.'' One of the most remarkable things about this book is that it was written, and published, in English not Latin. "I am at the last contented that it should come forth in English," he wrote resignedly, "Not that I think it worthy either of my labour or the publique view, but to satisfy their importunity who not understand the Latin yet were at the charge to buy the instrument."<ref name=who/> It was a manual not for cloistered university fellows but for sailors and surveyors in real world.
+which is logically equivalent to the [[Contraposition|contrapositive]],<ref>{{Cite web|url=http://www.math.umaine.edu/~farlow/sec42.pdf|title=Injections, Surjections, and Bijections|last=Farlow|first=S. J.|author-link= Stanley Farlow |website=math.umaine.edu|access-date=2019-12-06}}</ref><math display="block">\forall a, b \in X, \;\; a \neq b \Rightarrow f(a) \neq f(b).</math>
-There is reason to believe that Gunter was the first to discover (in 1622 or 1625) that the magnetic needle does not retain the same [[magnetic declination|declination]] in the same place at all times. By desire of [[James I of England|James I]] he published in 1624 ''The Description and Use of His Majesties Dials in Whitehall Garden'', the only one of his works which has not been reprinted. He coined the terms [[cosine]] and [[cotangent]], and he suggested to [[Henry Briggs (mathematician)|Henry Briggs]], his friend and colleague, the use of the arithmetical complement (see Briggs ''Arithmetica Logarithmica'', cap. xv).<ref name="EB1911"/> His practical inventions are briefly noted below:
-== Penemuan ==
-=== Gunter's chain ===
-Gunter's interest in geometry led him to develop a method of land surveying using triangulation. Linear measurements could be taken between topographical features such as corners of a field, and using triangulation the field or other area could be plotted on a plane, and its area calculated. A chain {{convert|66|ft}} long, with intermediate measurements indicated, was chosen for the purpose, and is called [[Gunter's chain]].
-The length of the chain chosen, {{convert|66|ft}}, being called a [[Chain (unit)|chain]] gives a unit easily converted to area.<ref>{{cite web |title=Gunter biography |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Gunter.html |website=www-history.mcs.st-andrews.ac.uk |access-date=21 July 2018}}</ref> Therefore, a parcel of 10 square chains gives 1 acre. The area of any parcel measured in chains will thereby be easily calculated.
-[[File:Table of Trigonometry, Cyclopaedia, Volume 2.jpg|thumb|Table of Trigonometry, from the 1728 ''[[Cyclopaedia, or an Universal Dictionary of Arts and Sciences|Cyclopaedia]]'', Volume 2 featuring a Gunter's scale]]
-=== Gunter's quadrant ===
-Gunter's quadrant is an instrument made of wood, brass or other substance, containing a kind of stereographic projection of the sphere on the plane of the equinoctial, the eye being supposed to be placed in one of the poles, so that the tropic, ecliptic, and horizon form the arcs of circles, but the hour circles are other curves, drawn by means of several altitudes of the sun for some particular latitude every year. This instrument is used to find the hour of the day, the sun's [[azimuth]], etc., and other common problems of the sphere or globe, and also to take the altitude of an object in degrees.<ref name="EB1911"/>
-A rare Gunter quadrant, made by Henry Sutton and dated 1657, can be described as follows: It is a conveniently sized and high-performance instrument that has two pin-hole sights, and the plumb line is inserted at the vertex. The front side is designed as a Gunter quadrant and the rear side as a trigonometric quadrant. The side with the astrolabe has hour lines, a calendar, zodiacs, star positions, astrolabe projections, and a vertical dial. The side with the geometric quadrants features several trigonometric functions, rules, a shadow quadrant, and the chorden line.<ref>Ralf Kern: ''Wissenschaftliche Instrumente in ihrer Zeit. Band 2: Vom Compendium zum Einzelinstrument''. Cologne, 2010; p. 205.</ref>
-=== Gunter's scale ===
-Gunter's scale or Gunter's rule, generally called the "Gunter" by seamen, is a large plane scale, usually {{convert|2|ft|mm}} long by about 1½ inches broad (40&nbsp;mm), engraved with various scales, or lines. On one side are placed the natural lines (as the line of chords, the line of [[sine]]s, [[tangent (trigonometric function)|tangent]]s, [[Rhumb line|rhumb]]s, etc.), and on the other side the corresponding artificial or logarithmic ones. By means of this instrument questions in [[navigation]], [[trigonometry]], etc., are solved with the aid of a pair of compasses.<ref name="EB1911"/>  It is a predecessor of the [[Slide rule#History|slide rule]], a calculating aid used from the 17th century until the 1970s.
-{{anchor|Gunter's line}}''Gunter's line'', or ''line of numbers'' refers to the logarithmically divided scale, like the most common scales used on slide rules for multiplication and division.
-=== Gunter rig ===
-A sail rig which resembles a gaff rig, with the gaff nearly vertical, is called a [[Gunter rig]], or "''sliding gunter"'' from its resemblance to a Gunter's rule.
-==See also==
-* [[Gresham Professor of Astronomy]]
-* [[History of geomagnetism]]
-==Notes==
-{{notelist}}
-==References==
-{{More footnotes|date=July 2019}}
-{{Reflist}}
-==External links==
-* {{MacTutor Biography|id=Gunter}}
-* [http://galileo.rice.edu/Catalog/NewFiles/gunter.html Galileo Project page]
-* [http://www.geoastro.de/gunter/ Gunter's Quadrant applet]
-{{Authority control}}
-{{DEFAULTSORT:Gunter, Edmund}}
-[[Category:1581 births]]
-[[Category:1626 deaths]]
-[[Category:English people of Welsh descent]]
-[[Category:17th-century English mathematicians]]
-[[Category:People educated at Westminster School, London]]
-[[Category:Alumni of Christ Church, Oxford]]
-[[Category:Fellows of Christ Church, Oxford]]
-[[Category:British scientific instrument makers]]
-[[Category:Professors of Gresham College]]
-[[Category:17th-century English Anglican priests]]

Fungsi
$x \mapsto f (x)$
Contoh domain dan kodomain fungsi
$X$ → $\mathbb {B}$ , $\mathbb {B}$ → $X$ , $\mathbb {B} ^{n}$ → $X$ $X$ → $\mathbb {Z}$ , $\mathbb {Z}$ → $X$ $X$ → $\mathbb {R}$ , $\mathbb {R}$ → $X$ , $\mathbb {R} ^{n}$ → $X$ $X$ → $\mathbb {C}$ , $\mathbb {C}$ → $X$ , $\mathbb {C} ^{n}$ → $X$
Kelas/sifat
Konstan Identitas Linear Polinomial Rasional Aljabar Analitik Mulus Kontinu Terukurkan Injektif Surjektif Bijektif
Konstruksi
Pembatasan Komposisi λ Invers
Perumuman
Parsial Bernilai banyak Implisit
l b s