Functions:Definition - Revision history

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A function (or "mapping") is a relationship between two sets <math> A </math> and <math> B </math> that maps each input <math> a\in A </math> to exactly one output <math> b\in B </math>. A function <math> f </math> that maps elements of the set <math> A </math> to elements in the set <math> B </math> is denoted as <math> f: A\to B </math>, where <math> A </math> is the domain of <math> f </math> and <math> B </math> is the codomain. We can also think of a function <math> f: A\to B </math> as a set of ordered pairs <math> (a, b) </math>, <math> a\in A </math> and <math> b\in B </math>, such that each element <math> a </math> is paired with exactly one element <math> b </math>. If a function <math> f: A\to B </math> maps an input <math> a </math> to an output <math> b </math>, we can write that <math> f(a) = b </math>.

Note: a function cannot map one input to more than one output, but it can map more than one input to the same output. For example, let <math> A = \{a_1, a_2\} </math> and <math> B = \{b_1, b_2\} </math>, <math> a_1\neq a_2 </math> and <math> b_1\neq b_2 </math>. If <math> f: A\to B </math> is a relation such that <math> f(a_1) = b_1 </math> and <math> f(a_1) = b_2 </math>, then <math> f </math> is NOT a function. However, a relation <math> g: A\to B </math> such that <math> g(a_1) = b_1 </math> and <math> g(a_2) = b_1 </math> IS a valid function.

Examples:
* Let <math> A = \{1, 2, 3, 4, 5\} </math> and <math> B = \{-1, 3, 4\} </math>, and let <math> f: A\to B </math> such that <math> f(1) = -1 </math>, <math> f(2) = 3 </math>, <math> f(3) = 4 </math>, <math> f(4) = 3 </math>, and <math> f(5) = -1 </math>. Since each element of the domain maps to exactly one element (that is, there is no <math> f(a) = b_1 </math> and <math> f(a) = b_2 </math> such that <math> b_1 \neq b_2 </math>), <math> f </math> is a function.
* For sets <math> A </math> and <math> B </math> in the previous example, let <math> g: A\to B </math> be a relation such that <math> g(1) = 1 </math>, <math> g(1) = 3 </math>, <math> g(2) = 4 </math>, <math> g(4) = 4 </math>, and <math> g(5) = 4 </math>. Since <math> g </math> maps the input <math> 1 </math> to two distinct outputs, this relation is NOT a valid function.
* Let <math> f: \N\to\N </math> such that <math> f(n) = n + 1 </math>. This is a function, since each <math> n\in\N </math> maps to exactly one element <math> n+1\in\N </math>.
* Let <math> g: \N\to\Z </math> such that |g(n)| = n. This is not a valid function, since for <math> n\in\N </math>, <math> g(n) <math> can equal both <math> n </math> and <math> -n </math>, and <math> n\neq -n </math> for <math> n\neq 0 </math>.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 129-140
* [https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/06%3A_Functions/6.02%3A_De%EF%AC%81nition_of_Functions Definition of Functions], Mathematics LibreTexts

@@ Line 1: / Line 1: @@
-[[File:Arrow diagram of a function (non-injective and non-surjective).svg|thumb|Example of an arrow diagram of a function <math> f:A\to B </math>]]
+== Functions ==
-A function (or "mapping") is a relationship between two sets <math> A </math> and <math> B </math> that maps each input <math> a\in A </math> to exactly one output <math> b\in B </math>. A function <math> f </math> that maps elements of the set <math> A </math> to elements in the set <math> B </math> is denoted as <math> f: A\to B </math>, where <math> A </math> is the domain of <math> f </math> and <math> B </math> is the codomain. We can also think of a function <math> f: A\to B </math> as a set of ordered pairs <math> (a, b) </math>, <math> a\in A </math> and <math> b\in B </math>, such that each element <math> a </math> is paired with exactly one element <math> b </math>. If a function <math> f: A\to B </math> maps an input <math> a </math> to an output <math> b </math>, we can write that <math> f(a) = b </math>. For finite, reasonably small sets, we can depict a function graphically (see image).
+{| align="right"
-Note: a function cannot map one input to more than one output, but it can map more than one input to the same output. For example, let <math> A = \{a_1, a_2\} </math> and <math> B = \{b_1, b_2\} </math>, <math> a_1\neq a_2 </math> and <math> b_1\neq b_2 </math>. If <math> f: A\to B </math> is a relation such that <math> f(a_1) = b_1 </math> and <math> f(a_1) = b_2 </math>, then <math> f </math> is NOT a function. However, a relation <math> g: A\to B </math> such that <math> g(a_1) = b_1 </math> and <math> g(a_2) = b_1 </math> IS a valid function.
+For example, if we write (define) a function as:
-===Examples===
+This function f maps numbers to their squares.
-* For sets <math> A </math> and <math> B </math> in the previous example, let <math> g: A\to B </math> be a relation such that <math> g(1) = 1 </math>, <math> g(1) = 3 </math>, <math> g(2) = 4 </math>, <math> g(4) = 4 </math>, and <math> g(5) = 4 </math>. Since <math> g </math> maps the input <math> 1 </math> to two distinct outputs, this relation is NOT a valid function.
+=== Range and codomain ===
-* Let <math> f: \N\to\N </math> such that <math> f(n) = n + 1 </math>. This is a function, since each <math> n\in\N </math> maps to exactly one element <math> n+1\in\N </math>.
+which forms a née of f is usually a subset of a larger set. This set is known as the ''codomain'' of a function. For example, with the function f(''x'')=cos ''x'', the range of f is [-1,1], '''but''' the codomain is the set of real numbers.
-* Let <math> g: \N\to\Z </math> such that <math> |g(n)| = n </math>. This is not a valid function, since for <math> n\in\N </math>, <math> g(n) </math> can equal both <math> n </math> and <math> -n </math>, and <math> n\neq -n </math> for <math> n\neq 0 </math>.
+=== Notations ===
 ==Resources==
 * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 129-140
 * [https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/06%3A_Functions/6.02%3A_De%EF%AC%81nition_of_Functions Definition of Functions], Mathematics LibreTexts

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	Note: a function cannot map one input to more than one output, but it can map more than one input to the same output. For example, let <math> A = \{a_1, a_2\} </math> and <math> B = \{b_1, b_2\} </math>, <math> a_1\neq a_2 </math> and <math> b_1\neq b_2 </math>. If <math> f: A\to B </math> is a relation such that <math> f(a_1) = b_1 </math> and <math> f(a_1) = b_2 </math>, then <math> f </math> is NOT a function. However, a relation <math> g: A\to B </math> such that <math> g(a_1) = b_1 </math> and <math> g(a_2) = b_1 </math> IS a valid function.		Note: a function cannot map one input to more than one output, but it can map more than one input to the same output. For example, let <math> A = \{a_1, a_2\} </math> and <math> B = \{b_1, b_2\} </math>, <math> a_1\neq a_2 </math> and <math> b_1\neq b_2 </math>. If <math> f: A\to B </math> is a relation such that <math> f(a_1) = b_1 </math> and <math> f(a_1) = b_2 </math>, then <math> f </math> is NOT a function. However, a relation <math> g: A\to B </math> such that <math> g(a_1) = b_1 </math> and <math> g(a_2) = b_1 </math> IS a valid function.
−

	===Examples===		===Examples===

← Older revision		Revision as of 19:08, 27 September 2021
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−	Examples:	+	===Examples===
	* Let <math> A = \{1, 2, 3, 4, 5\} </math> and <math> B = \{-1, 3, 4\} </math>, and let <math> f: A\to B </math> such that <math> f(1) = -1 </math>, <math> f(2) = 3 </math>, <math> f(3) = 4 </math>, <math> f(4) = 3 </math>, and <math> f(5) = -1 </math>. Since each element of the domain maps to exactly one element (that is, there is no <math> f(a) = b_1 </math> and <math> f(a) = b_2 </math> such that <math> b_1 \neq b_2 </math>), <math> f </math> is a function.		* Let <math> A = \{1, 2, 3, 4, 5\} </math> and <math> B = \{-1, 3, 4\} </math>, and let <math> f: A\to B </math> such that <math> f(1) = -1 </math>, <math> f(2) = 3 </math>, <math> f(3) = 4 </math>, <math> f(4) = 3 </math>, and <math> f(5) = -1 </math>. Since each element of the domain maps to exactly one element (that is, there is no <math> f(a) = b_1 </math> and <math> f(a) = b_2 </math> such that <math> b_1 \neq b_2 </math>), <math> f </math> is a function.

← Older revision		Revision as of 18:15, 27 September 2021
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−	[[File:Example of an arrow diagram of a function <math> f:A\to B </math>~~.svg\|Arrow_diagram_of_a_function_(non-injective_and_non-surjective)~~]]	+	[[File:Arrow diagram of a function (non-injective and non-surjective).svg\|thumb\|Example of an arrow diagram of a function <math> f:A\to B </math>]]

	A function (or "mapping") is a relationship between two sets <math> A </math> and <math> B </math> that maps each input <math> a\in A </math> to exactly one output <math> b\in B </math>. A function <math> f </math> that maps elements of the set <math> A </math> to elements in the set <math> B </math> is denoted as <math> f: A\to B </math>, where <math> A </math> is the domain of <math> f </math> and <math> B </math> is the codomain. We can also think of a function <math> f: A\to B </math> as a set of ordered pairs <math> (a, b) </math>, <math> a\in A </math> and <math> b\in B </math>, such that each element <math> a </math> is paired with exactly one element <math> b </math>. If a function <math> f: A\to B </math> maps an input <math> a </math> to an output <math> b </math>, we can write that <math> f(a) = b </math>. For finite, reasonably small sets, we can depict a function graphically (see image).		A function (or "mapping") is a relationship between two sets <math> A </math> and <math> B </math> that maps each input <math> a\in A </math> to exactly one output <math> b\in B </math>. A function <math> f </math> that maps elements of the set <math> A </math> to elements in the set <math> B </math> is denoted as <math> f: A\to B </math>, where <math> A </math> is the domain of <math> f </math> and <math> B </math> is the codomain. We can also think of a function <math> f: A\to B </math> as a set of ordered pairs <math> (a, b) </math>, <math> a\in A </math> and <math> b\in B </math>, such that each element <math> a </math> is paired with exactly one element <math> b </math>. If a function <math> f: A\to B </math> maps an input <math> a </math> to an output <math> b </math>, we can write that <math> f(a) = b </math>. For finite, reasonably small sets, we can depict a function graphically (see image).