Functions:Bijective - Revision history

Khanh at 21:29, 11 November 2021

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Lila at 21:10, 27 September 2021

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Lila at 21:09, 27 September 2021

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Lila at 21:08, 27 September 2021

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Lila at 20:52, 27 September 2021

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Lila: Created page with "A function is bijective if it is both injective and surjective. That is, a bijective function maps each element of the domain to a distinct element in the codomain, and every..."

2021-09-27T20:44:25Z

Created page with "A function is bijective if it is both injective and surjective. That is, a bijective function maps each element of the domain to a distinct element in the codomain, and every..."

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A function is bijective if it is both injective and surjective. That is, a bijective function maps each element of the domain to a distinct element in the codomain, and every element in the codomain is mapped to by exactly one element of the domain.

[[File:Injective, Surjective, Bijective.svg|Injective, Surjective, and Bijective arrow diagrams]]

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 In mathematics, a '''bijection''', '''bijective function''', '''one-to-one correspondence''', or '''invertible function''', is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function ''f'': ''X'' → ''Y'' is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures).
-<div style="text-align: center;"><gallery widths="250">
+<div style="text-align: center;"><gallery widths="200">
 Image:Injection.svg|An injective non-surjective function (injection, not a bijection)
 Image:Bijection.svg|An injective surjective function ('''bijection''')

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 * <math> f:A\to B, A = \{a, b, c\}, B = \{1, 2, 3, 4, 5\}</math> such that <math> f(a) = 1 </math>, <math> f(b) = 2 </math>, and  <math> f(c) = 3 </math> (each input has a unique output, but not all elements of the codomain are mapped to)
 * <math> f:\N\to\N, f(n) = n^2 </math>
-* <math> f:{x\in\R, x \ge 0}\to\R, f(x) = x^2 + 100 </math>
+* <math> f:\{x\in\R, x \ge 0\}\to\R, f(x) = x^2 + 100 </math>
 Surjective, not injective:
 * <math> f:A\to B, A = \{a, b, c, d\}, B = \{1, 2, 3\} </math> such that <math> f(a) = 1 </math>, <math> f(b) = 1 </math>, <math> f(c) = 2 </math>, and <math> f(d) = 3 </math>
-* <math> f:\R\to{x\in\R, x \ge 0}, f(n) = |x| </math>
+* <math> f:\R\to\{x\in\R, x \ge 0\}, f(n) = |x| </math>
 * <math> f:\R\to\R, f(x) = x^2 + 2x + 1 </math> (<math> -2 </math> and <math>0</math> map to the same output, so not injective; range is <math> (-\infty, \infty) = \R </math>, so surjective)

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 * <math> f:A\to B, A = \{a, b, c, d\}, B = \{1, 2, 3\} </math> such that <math> f(a) = 1 </math>, <math> f(b) = 1 </math>, <math> f(c) = 2 </math>, and <math> f(d) = 3 </math>
 * <math> f:\R\to{x\in\R, x \ge 0}, f(n) = |x| </math>
-* <math> f:\R\to\R, f(x) = x^2 + 2x + 1 </math> (-2 and 0 map to the same output, so not injective; range is <math> (-\infty, \infty) = \R </math>, so surjective)
+* <math> f:\R\to\R, f(x) = x^2 + 2x + 1 </math> (<math> -2 </math> and <math>0</math> map to the same output, so not injective; range is <math> (-\infty, \infty) = \R </math>, so surjective)
 Bijections:

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 [[File:Injective, Surjective, Bijective.svg|Injective, Surjective, and Bijective arrow diagrams]]
-Examples of bijective functions:
+===Examples===
 * <math> f:A\to B, A = \{a, b, c\}, B = \{1, 2, 3\}</math> such that <math> f(a) = 1 </math>, <math> f(b) = 2 </math>, and  <math> f(c) = 3 </math>
 * <math> f:\R\to\R, f(x) = 3x + 5 </math>

← Older revision		Revision as of 20:52, 27 September 2021
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	[[File:Injective, Surjective, Bijective.svg\|Injective, Surjective, and Bijective arrow diagrams]]		[[File:Injective, Surjective, Bijective.svg\|Injective, Surjective, and Bijective arrow diagrams]]
		+
		+	Examples of bijective functions:
		+	* <math> f:A\to B, A = \{a, b, c\}, B = \{1, 2, 3\}</math> such that <math> f(a) = 1 </math>, <math> f(b) = 2 </math>, and <math> f(c) = 3 </math>
		+	* <math> f:\R\to\R, f(x) = 3x + 5 </math>
		+	* <math> f:\R\to\R, f(x) = x^3 </math>
		+
		+	==Resources==
		+	* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 154-164