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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.

**^**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, vol. 154, Springer, p. 8, ISBN 9780387943695.**^**Jech, Thomas (2003),*Set Theory: The Third Millennium Edition*, Springer Monographs in Mathematics, Springer, p. 34, ISBN 9783540440857.