Editing Exact sequence (section)

===Simple cases===
To understand the definition, it is helpful to consider relatively simple cases where the sequence is of group homomorphisms, is finite, and begins or ends with the [[trivial group]]. Traditionally, this, along with the single identity element, is denoted 0 (additive notation, usually when the groups are abelian), or denoted 1 (multiplicative notation).

* Consider the sequence 0 → ''A'' → ''B''. The image of the leftmost map is 0. Therefore the sequence is exact if and only if the rightmost map (from ''A'' to ''B'') has kernel {0}; that is, if and only if that map is a [[monomorphism]] (injective, or one-to-one).
* Consider the dual sequence ''B'' → ''C'' → 0. The kernel of the rightmost map is ''C''. Therefore the sequence is exact if and only if the image of the leftmost map (from ''B'' to ''C'') is all of ''C''; that is, if and only if that map is an [[epimorphism]] (surjective, or onto).
* Therefore, the sequence 0 → ''X'' → ''Y'' → 0 is exact if and only if the map from ''X'' to ''Y'' is both a monomorphism and epimorphism (that is, a [[bimorphism]]), and so usually an [[isomorphism]] from ''X'' to ''Y'' (this always holds in [[exact categories]] like '''Set''').