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Any homomorphism f : X -> Y defines an equivalence relation on X by a ~ b iff f(a) = f(b). The quotient set X / ~ can then be given an object-structure in a natural way, e.g. [x] * [y] =[x * y]. In that case the image of X is necessarily isomorphic to X / ~. Note in some cases (e.g. groups) a single equivalence class Y suffices to specify the structure of the quotient, so we write it X / Y. |
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Any homomorphism f : X -> Y defines an equivalence relation on X by a ~ b iff f(a) = f(b). The quotient set X / ~ can then be given an object-structure in a natural way, e.g. [x] * [y] =[x * y]. In that case the image of X is necessarily isomorphic to X / ~. Note in some cases (e.g. groups) a single equivalence class U suffices to specify the structure of the quotient, so we write it X / U. |
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