Difference between revisions of "Injection"
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: an '''injective function''' or an '''injection''' | : an '''injective function''' or an '''injection''' | ||
if and only if | if and only if | ||
− | : implies $ f(a_1) | + | : a_1 \not= a_1 \in A implies $ f(a_1) \ne f(a_2) , or equivalently f(a_1) = f(a_2) implies a_1 = a_2 $. |
A special case is the ''inclusion'' function defined on a subset A \subset B by f(a)=a . | A special case is the ''inclusion'' function defined on a subset A \subset B by f(a)=a . |
Revision as of 07:11, 16 February 2012
A function (or mapping) is called injective if distinct arguments have distinct images.
In other words, a function f : A \to B from a set A to a set B is
- an injective function or an injection
if and only if
- a_1 \not= a_1 \in A implies f(a_1) \ne f(a_2) , or equivalently f(a_1) = f(a_2) implies a_1 = a_2 .
A special case is the inclusion function defined on a subset A \subset B by f(a)=a .
An injective homomorphism is called monomorphism.
Injective mappings that are compatible with the underlying structure are often called embedding.
A function that is both injective and surjective is called bijective (or, if domain and range coincide, in some contexts, a permutation).
A function f is injective if and only if f \circ g = f \circ h then g=h . In category theory, this property is used to define injective morphisms.
Injection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injection&oldid=21083