Preimage
the original figure in a transformation
In mathematics, the preimage of a set is the set of all input values that produce a given output value when passed through a function. In other words, if we have a function f: X → Y, and a subset A of Y, then the preimage of A is the set of all x values in X such that f(x) belongs to A.
The preimage of A is denoted by f^(-1)(A). It is important to note that the preimage of a set may not be unique, since different input values may produce the same output value.
For example, let f: R → R be the function defined by f(x) = x^2, and let A = {1, 4}. The preimage of A under f is given by f^(-1)(A) = {-2, -1, 1, 2}, since f(-2) = 4, f(-1) = 1, f(1) = 1, and f(2) = 4.
Preimages are especially important in areas of mathematics such as topology, where they are used to define open and closed sets, continuity, and other key concepts.
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