Homomorphism - Kernel of A Homomorphism

Kernel of A Homomorphism

Any homomorphism f : X → Y defines an equivalence relation ~ on X by a ~ b if and only if f(a) = f(b). The relation ~ is called the kernel of f. It is a congruence relation on X. The quotient set X/~ can then be given an object-structure in a natural way, i.e. * = . In that case the image of X in Y under the homomorphism f is necessarily isomorphic to X/~; this fact is one of the isomorphism theorems. Note in some cases (e.g. groups or rings), a single equivalence class K suffices to specify the structure of the quotient; so we can write it X/K. (X/K is usually read as "X mod K".) Also in these cases, it is K, rather than ~, that is called the kernel of f (cf. normal subgroup).