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injective function (Definition)

We say that a function $ f\colon A\to B$ is injective or one-to-one if $ f(x)=f(y)$ implies $ x=y$, or equivalently, whenever $ x\neq y$, then $ f(x)\neq f(y)$.

Properties

  1. Suppose $ A,B,C$ are sets and $ f\colon A\to B$, $ g\colon B\to C$ are injective functions. Then the composition $ g\circ f$ is an injection.
  2. Suppose $ f\colon A\to B$ is an injection, and $ C\subseteq A$. Then the restriction $ f\vert _C\colon C\to B$ is an injection.

For a list of other properties of injective functions, see [1].

Bibliography

1
Wikipedia, article on Injective function.



"injective function" is owned by drini. [ full author list (3) | owner history (1) ]
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See Also: bijection, function, surjective

Other names:  one-to-one, injection, embedding, injective

Attachments:
an injection between two finite sets of the same cardinality is bijective (Theorem) by alozano
properties of injective functions (Theorem) by rspuzio
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Cross-references: restriction, composition, implies, function
There are 178 references to this entry.

This is version 11 of injective function, born on 2001-10-20, modified 2007-02-19.
Object id is 429, canonical name is InjectiveFunction.
Accessed 39760 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)
Pending Errata and Addenda
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forum policy
Wikipedia has a bad reputation by zorba on 2005-12-09 16:31:02
Because I saw the citation in this entry to wikipedia
I forward an e-mail which was circulated around and I received
from my cousin Demetrios who lives in New York.

" Wikipedia has a bad reputation in America. Recently
" in an article in the news on TV, Yahoo news, press, and
" even at the bank's display it was said wikipedia
" uses false accusations with no proof, undocumented stuff, biased
" articles.
" People I know consider it a junk. Read the article below
" got on December 5th 2005 from
" http://news.yahoo.com/s/ap/20051205/ap_on_hi_te/wikipedia_rules


Online Encyclopedia Tightens Rules By DAN GOODIN,
Associated Press Writer
2 hours, 35 minutes ago

SAN FRANCISCO - Wikipedia, the online encyclopedia
that allows anyone to contribute articles, is
tightening its rules for submitting entries following
the disclosure that it ran a piece falsely implicating
a man in the Kennedy assassinations.

Wikipedia will now require users to register before
they can create articles, Jimmy Wales, founder of the
St. Petersburg, Fla.-based Web site, said Monday.

The change comes less than a week after John
Seigenthaler Sr., who was Robert Kennedy's
administrative assistant in the early 1960s, wrote an
op-ed article revealing that Wikipedia had run a
biography claiming Seigenthaler had been suspected in
the assassinations of the former Attorney General and
his brother, President John F. Kennedy.

Wikipedia, which on Monday offered more than 850,000
articles in English, has grown into a storehouse of
pieces on topics ranging from medieval art to nano
technology. The volume of content is possible because
the site relies on volunteers, including many experts
in their fields, to submit entries and edit previously
submitted articles.

The Web site hopes that the registration requirement
will limit the number of stories being created, Wales
said.

"What we're hopeful to see is that by slowing that
down to 1,500 a day from several thousand, the people
who are monitoring this will have more ability to
improve the quality," Wales said Monday. "In many
cases the types of things we see going on are impulse
vandalism."

Wikipedia visitors will still be able to edit content
already posted without registering. It takes 15 to 20
seconds to create an account on the Web site, and an
e-mail address is not required.

Seigenthaler, a former newspaper editor at the
Tennessean in Nashville, Tenn., and founder of the
Freedom Forum First Amendment Center at Vanderbilt
University, said that following his op-ed piece in USA
Today the biography of him was changed to remove the
false accusations.

But Seigenthaler said the current entry on Monday
still got some facts wrong, apparently because
volunteers are confusing him with his son, John
Seigenthaler Jr., a journalist with NBC News.

Also disturbing is a section of his biography that
tracks changes made to the article, Seigenthaler, Sr.
said. Entries in that history section label him a
"Nazi" and say other "really vicious, venomous,
salacious homophobic things about me," he said.

Wales said those comments would be removed.

For 132 days, Seigenthaler said the biography of him
falsely claimed that "for a brief time, he was thought
to have been directly involved in the Kennedy
assassinations of both John, and his brother, Bobby."

The biography also falsely stated that he had lived in
the Soviet Union from 1971 to 1984.

Seigenthaler said he wasn't convinced the new
registration requirement would stop the practice of
vandals posting content that is slanderous or
knowingly incorrect. Wikipedia will either have to fix
the problem or will lose whatever credibility it still
has, he said.

"The marketplace of ideas ultimately will take care of
the problem," Seigenthaler said. "In the meantime,
what happens to people like me?"
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