Difference between revisions of "Functions:Bijective"
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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: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> (-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: | Bijections: | ||
Revision as of 15:09, 27 September 2021
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.
Examples
Injective, not surjective:
- 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, A = \{a, b, c\}, B = \{1, 2, 3, 4, 5\}} such that , , and (each input has a unique output, but not all elements of the codomain are mapped to)
Surjective, not injective:
- 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(c) = 2 } , and 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(d) = 3 }
- 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:\R\to{x\in\R, x \ge 0}, f(n) = |x| }
- 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:\R\to\R, f(x) = x^2 + 2x + 1 } (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 -2 } and 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 0} map to the same output, so not injective; range is 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 (-\infty, \infty) = \R } , so surjective)
Bijections:
- 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, A = \{a, b, c\}, B = \{1, 2, 3\}} 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) = 1 } , 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(b) = 2 } , and 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(c) = 3 }
- 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:\R\to\R, f(x) = 3x + 5 }
- 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:\R\to\R, f(x) = x^3 }
Resources
- Course Textbook, pages 154-164
