preimage
In mathematics, a preimage refers to the original set of elements in the domain of a function that gets mapped to another set of elements in the range
In mathematics, a preimage refers to the original set of elements in the domain of a function that gets mapped to another set of elements in the range.
Let’s say we have a function f that maps elements from set A to set B. The preimage of an element y in set B is the set of all elements x in set A such that f(x) = y. The preimage is denoted as f^(-1)(y), where the (-1) superscript does not denote the inverse function but represents the preimage operation.
To visualize this, let’s consider a simple function f(x) = x^2, which maps elements from set A (real numbers) to set B (non-negative real numbers). If we take the preimage of the element 4, denoted as f^(-1)(4), we need to find all the elements in set A that get mapped to 4.
Since f(x) = x^2, we can solve for x in the equation x^2 = 4 by taking the square root of both sides. This gives us two possible solutions: x = 2 or x = -2. Thus, the preimage of 4 is {2, -2}.
In general, the preimage of an element y under a function can be a single element, multiple elements, or even an empty set, depending on the function and the specific element in the range. It is important to note that the preimage is not necessarily unique, as different elements in the domain can map to the same element in the range.
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