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### projection (set theory)

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ARTICLE ORIGINS projection (set theory)
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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 jth projection map, written mathrm{proj}_{j}, that takes an element vec{x} = (x_1, ldots, x_j, ldots, x_k) of the Cartesian product (X_1 times cdots times X_j times cdots times X_k) to the value mathrm{proj}_{j}(vec{x}) = x_j.{{citation|title=Naive Set Theory|series=Undergraduate Texts in Mathematics|first=P. R.|last=Halmos|authorlink=Paul Halmos|publisher=Springer|year=1960|isbn=9780387900926|page=32|url=https://books.google.com/books?id=x6cZBQ9qtgoC&pg=PA32}}.
• A function that sends an element x to its equivalence class under a specified equivalence relation E,{{citation|title=An Introduction to Analysis|volume=154|series=Graduate Texts in Mathematics|first1=Arlen|last1=Brown|first2=Carl M.|last2=Pearcy|publisher=Springer|year=1995|isbn=9780387943695|page=8|url=https://books.google.com/books?id=Y2Mwck8Q9A4C&pg=PA8}}. or, equivalently, a surjection from a set to another set.{{citation|title=Set Theory: The Third Millennium Edition|series=Springer Monographs in Mathematics|first=Thomas|last=Jech|publisher=Springer|year=2003|isbn=9783540440857|page=34|url=https://books.google.com/books?id=WTAl997XDb4C&pg=PA34}}. 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

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