Difference between revisions of "Injection"
From Maths
(Created page with "An injective function is 1:1, but not nessasarally onto. For <math>f:X\rightarrow Y</math> every element of <math>X</math> is mapped to an element of <math>Y</...") |
m |
||
| Line 1: | Line 1: | ||
An injective function is 1:1, but not nessasarally [[Surjection|onto]]. | An injective function is 1:1, but not nessasarally [[Surjection|onto]]. | ||
| + | ==Definition== | ||
| + | For a [[Function|function]] <math>f:X\rightarrow Y</math> every element of <math>X</math> is mapped to an element of <math>Y</math> and no two distinct things in <math>X</math> are mapped to the same thing in <math>Y</math>. That is: | ||
| + | * <math>\forall x_1,x_2\in X[f(x_1)=f(x_2)\implies x_1=x_2]</math> | ||
| + | Or equivalently: | ||
| + | * <math>\forall x_1,x_2\in X[x_1\ne x_2\implies f(x_1)=f(x_2)]</math> (the [[Contrapositive|contrapositive]] of the above) | ||
| − | + | ==Notes== | |
| + | The cardinality of the inverse of an element <math>y\in Y</math> may be no more than 1; that is it may be zero, in contrast to a bijection where the cardinality is always 1 (and thus we take the singleton set <math>f^{-1}(y)=\{x\}</math> as the value it contains) | ||
| − | + | {{Todo|Find reference - should be easy!}} | |
| − | + | ==See also== | |
| + | * [[Bijection]] | ||
| + | * [[Surjection]] | ||
| + | * [[Function]] | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
| − | {{Definition}} | + | {{Definition|Set Theory}} |
Revision as of 12:58, 16 June 2015
An injective function is 1:1, but not nessasarally onto.
Contents
Definition
For a function [math]f:X\rightarrow Y[/math] every element of [math]X[/math] is mapped to an element of [math]Y[/math] and no two distinct things in [math]X[/math] are mapped to the same thing in [math]Y[/math]. That is:
- [math]\forall x_1,x_2\in X[f(x_1)=f(x_2)\implies x_1=x_2][/math]
Or equivalently:
- [math]\forall x_1,x_2\in X[x_1\ne x_2\implies f(x_1)=f(x_2)][/math] (the contrapositive of the above)
Notes
The cardinality of the inverse of an element [math]y\in Y[/math] may be no more than 1; that is it may be zero, in contrast to a bijection where the cardinality is always 1 (and thus we take the singleton set [math]f^{-1}(y)=\{x\}[/math] as the value it contains)
TODO: Find reference - should be easy!
