Functions:Injective - Revision history

Khanh at 22:34, 7 November 2021

2021-11-07T22:34:34Z

Khanh at 22:27, 7 November 2021

2021-11-07T22:27:49Z

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Lila at 20:15, 27 September 2021

2021-09-27T20:15:08Z

Lila: Created page with "A function $f: A\to B$ is injective, or "one-to-one", if for all $a_1, a_2\in A$ , $a_1 \neq a_2$ implies that $f(a_1) \neq f(a_2)..."$

2021-09-27T19:57:15Z

Created page with "A function <math> f: A\to B </math> is injective, or "one-to-one", if for all <math> a_1, a_2\in A </math>, <math> a_1 \neq a_2 </math> implies that <math> f(a_1) \neq f(a_2)..."

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A function <math> f: A\to B </math> is injective, or "one-to-one", if for all <math> a_1, a_2\in A </math>, <math> a_1 \neq a_2 </math> implies that <math> f(a_1) \neq f(a_2) </math> (or for all <math> a_1, a_2\in A </math>, <math> f(a_1) = f(a_2) </math> implies that <math> a_1 = a_2 </math>). That is, a function is injective if each output is unique to a specific input, and no two distinct inputs map to the same output.

Examples:
* Let <math> A = \{1, 2, 3\} </math> and <math> B = \{1, 2, 3\} </math>, and let <math> f:A\to B </math> such that <math> f(1) = 2 </math>, <math> f(2) = 1 </math>, and <math> f(3) = 3 </math>. <math> f </math> is an injective function because each output of <math> f </math> is mapped to by exactly one input.

* Let <math> g:A\to B </math> such that <math> g(1) = 1 </math>, <math> g(2) = 1 </math>, and <math> g(3) = 2 </math>. <math> g </math> is not an injective function since <math> g(1) = g(2) </math>.

* <math> f:\R\to\R </math>, <math> f(x) = x </math> is an injective function, since <math> x_1 \neq x_2 \implies f(x_1) \neq f(x_2) </math> for all <math> x_1,x_2\in\R </math>.

* Let <math> f:\R\to\R </math>, <math> f(x) = x^2 </math>. This function is NOT injective because for <math> x \neq 0 </math>, <math> f(x) = x^2 = (-x)^2 = f(-x) </math>, but <math> x \neq -x </math>. For example, <math> 1\neq -1</math> while <math> f(1) = f(-1) = 1</math>, which conflicts with the definition of injectivity.

==Resources==
* [https://en.wikipedia.org/wiki/Injective_function Injective Function], Wikipedia
* [https://cnx.org/contents/ysm8oGY0@64.2:jJWptB8O@4/Function-types Function Types], OpenStax
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Proofs and Fundamentals: A First Course in Abstract Mathematics], pages 154-164

Also see [[Functions:Definition|functions]].

@@ Line 1: / Line 1: @@
 In mathematics, an '''injective function''' (also known as '''injection''', or '''one-to-one function''') is a function {{math|''f''}} that maps distinct elements to distinct elements; that is, ''f''(''x''<sub>1</sub>) = ''f''(''x''<sub>2</sub>) implies ''x''<sub>1</sub> = ''x''<sub>2</sub>. In other words, every element of the function's codomain is the image of ''at most'' one element of its domain. 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.
-<gallery widths="200">
+<gallery widths="150">
-|Image:Injection.svg|An '''injective''' non-surjective function (injection, not a bijection)
+Image:Injection.svg|An '''injective''' non-surjective function (injection, not a bijection)
-|Image:Bijection.svg|An '''injective''' surjective function (bijection)
+Image:Bijection.svg|An '''injective''' surjective function (bijection)
-|Image:Surjection.svg|A non-injective surjective function (surjection, not a bijection)
+Image:Surjection.svg|A non-injective surjective function (surjection, not a bijection)
-|Image:Not-Injection-Surjection.svg|A non-injective non-surjective function (also not a bijection)
+Image:Not-Injection-Surjection.svg|A non-injective non-surjective function (also not a bijection)
 </gallery>

@@ Line 13: / Line 13: @@
 * [https://en.wikipedia.org/wiki/Injective_function Injective Function], Wikipedia
 * [https://cnx.org/contents/ysm8oGY0@64.2:jJWptB8O@4/Function-types Function Types], OpenStax
-* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Proofs and Fundamentals: A First Course in Abstract Mathematics], pages 154-164
+* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 154-164
 Also see [[Functions:Definition|functions]].

Functions:Injective - Revision history

Khanh at 22:34, 7 November 2021

Khanh at 22:27, 7 November 2021

Lila at 20:15, 27 September 2021

Lila: Created page with "A function f: A\to B is injective, or "one-to-one", if for all a_1, a_2\in A , a_1 \neq a_2 implies that f(a_1) \neq f(a_2)..."

Lila: Created page with "A function $f: A\to B$ is injective, or "one-to-one", if for all $a_1, a_2\in A$ , $a_1 \neq a_2$ implies that $f(a_1) \neq f(a_2)..."$