# Projection (set theory)

In set theory, a **projection** is one of two closely related types of functions or operations, namely:

- A set-theoretic operation typified by the
*j*^{th}projection map, written , that takes an element of the Cartesian product to the value .[1] - A function that sends an element
*x*to its equivalence class under a specified equivalence relation*E*,[2] or, equivalently, a surjection from a set to another set.[3] The function from elements to equivalence classes is a surjection, and every surjection corresponds to an equivalence relation under which two elements are equivalent when they have the same image. The result of the mapping is written as [*x*] when*E*is understood, or written as [*x*]_{E}when it is necessary to make*E*explicit.

## References

- Halmos, P. R. (1960),
*Naive Set Theory*, Undergraduate Texts in Mathematics, Springer, p. 32, ISBN 9780387900926. - Brown, Arlen; Pearcy, Carl M. (1995),
*An Introduction to Analysis*, Graduate Texts in Mathematics,**154**, Springer, p. 8, ISBN 9780387943695. - Jech, Thomas (2003),
*Set Theory: The Third Millennium Edition*, Springer Monographs in Mathematics, Springer, p. 34, ISBN 9783540440857.

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