Difference between revisions of "One-to-one functions"
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| + | To make it simple, for the function <math>f(x)</math>, all of the possible <math>x</math> values constitute the domain, and all of the values <math>f(x)</math> (<math>y</math> on the x-y plane) constitute the range. To put it in more formal terms, a function <math>f</math> is a mapping of some element <math>a\in A</math>, called the domain, to exactly one element <math>b\in B</math>, called the range, such that <math>f:A\to B</math>. The image below should help explain the modern definition of a function: | ||
| + | [[File:Function_Definition.svg|alt=The image demonstrates a mapping of some element a (the circle) in A, the domain, to exactly one element b in B, the range.|thumb|<math>A</math> is the domain of the function while <math>B</math> is the range. This transformation from set <math>A</math> to <math>B</math> is an example of one-to-one function.]] | ||
| + | : A function is considered '''one-to-one''' if an element <math>a\in A</math> from domain <math>A</math> of function <math>f</math>, leads to exactly one element <math>b\in B</math> from range <math>B</math> of the function. By definition, since only one element <math>b</math> is mapped by function <math>f</math> from some element <math>a</math>, <math>f:A\to B</math> implies that there exists only one element <math>b</math> from the mapping. Therefore, there exists a one-to-one function because it complies with the definition of a function. This definition is similar to '''''Figure 1'''''. | ||
| + | |||
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/One-to-one%20functions/Esparza%201093%20Notes%201.7.pdf One-to-one functions]. Written notes created by Professor Esparza, UTSA. | * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/One-to-one%20functions/Esparza%201093%20Notes%201.7.pdf One-to-one functions]. Written notes created by Professor Esparza, UTSA. | ||
Revision as of 12:12, 5 October 2021
To make it simple, for the function , all of the possible values constitute the domain, and all of the values ( on the x-y plane) constitute the range. To put it in more formal terms, a function is a mapping of some element , called the domain, to exactly one element , called the range, such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:A\to B} . The image below should help explain the modern definition of a function:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
is the domain of the function while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}
is the range. This transformation from set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}
is an example of one-to-one function.
- A function is considered one-to-one if an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in A} from domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} of function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , leads to exactly one element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\in B} from range Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} of the function. By definition, since only one element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is mapped by function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} from some element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:A\to B} implies that there exists only one element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} from the mapping. Therefore, there exists a one-to-one function because it complies with the definition of a function. This definition is similar to Figure 1.
- One-to-one functions. Written notes created by Professor Esparza, UTSA.
