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 $\backslash mathrm\{proj\}\_j,$ that takes an element $\backslash vec\{x\}\; =\; (x\_1,\backslash \; \backslash dots,\backslash \; x\_j,\backslash \; \backslash dots,\backslash \; x\_k)$ of the Cartesian product $(X\_1\; \backslash times\; \backslash cdots\; \backslash times\; X\_j\; \backslash times\; \backslash cdots\; \backslash times\; X\_k)$ to the value $\backslash mathrm\{proj\}\_j(\backslash vec\{x\})\; =\; x\_j.${{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.