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A permutation is generally defined as a bijection on a non-empty set.

Given a permutation on a set W, a partial permutation is a bijection from one subset W_X of W to another subset W_Y of W.  W_X and W_Y are the domain and range, respectively, of the partial permutation.  Because a partial permutation is a bijection, W_X and W_Y must contain the same number of elements (or must be of the same cardinality, if they are infinite).  Note that a partial permutation is defined only in the context of a specific and previously defined permutation.  Generally speaking, a partial permutation is not a permutation, and indeed a partial permutation is a permutation only if its domain and range are the same set.