 TOPICS # Pre-Image

Let be a map between sets and . Let . Then the preimage of under is denoted by , and is the set of all elements of that map to elements in under . Thus (1)

One is not to be mislead by the notation into thinking of the preimage as having to do with an inverse of . The preimage is defined whether has an inverse or not. Note however that if does have an inverse, then the preimage is exactly the image of under the inverse map, thus justifying the perhaps slightly misleading notation.

For any , it is true that (2)

with equality occurring, if is surjective, and for any subset , it is true that (3)

with equality occurring if is injective.

Preimages occur in a variety of subjects, the most persistent of these being topology, where a map is continuous, by definition, if the preimage of every open set is open.

Image

This entry contributed by Rasmus Hedegaard

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Hedegaard, Rasmus. "Pre-Image." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Pre-Image.html