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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 {\displaystyle j} th projection map, written p r o j j , {\displaystyle \mathrm {proj} _{j},} that takes an element x → → = ( x 1 , … … , x j , … … , x k ) {\displaystyle {\vec {x}}=(x_{1},\ \dots ,\ x_{j},\ \dots ,\ x_{k})} of the Cartesian product ( X 1 × × ⋯ ⋯ × × X j × × ⋯ ⋯ × × X k ) {\displaystyle (X_{1}\times \cdots \times X_{j}\times \cdots \times X_{k})} to the value p r o j j ( x → → ) = x j . {\displaystyle \mathrm {proj} _{j}({\vec {x}})=x_{j}.} cite-ref-1[1]

• A function that sends an element x {\displaystyle x} to its equivalence class under a specified equivalence relation E , {\displaystyle E,} cite-ref-2[2] or, equivalently, a surjection from a set to another set.cite-ref-3[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 ] {\displaystyle [x]} when E {\displaystyle E} is understood, or written as [ x ] E {\displaystyle [x]_{E}} when it is necessary to make E {\displaystyle E} explicit.

• See also

• References

• Cartesian product – Mathematical set formed from two given sets

• Projection (mathematics) – Mapping equal to its square under mapping composition

• Projection (measure theory)

• Projection (linear algebra) – Idempotent linear transformation from a vector space to itself

• Projection (relational algebra) – Operation that restricts a relation to a specified set of attributes

• Relation (mathematics) – Relationship between two sets, defined by a set of ordered pairs

cite-note-11. ↑ citerefhalmos1960Halmos, P. R. (1960), Naive Set Theory, Undergraduate Texts in Mathematics, Springer, p. 32, ISBN 9780387900926 {{citation}}: ISBN / Date incompatibility (help).

cite-note-22. ↑ citerefbrownpearcy1995Brown, Arlen; Pearcy, Carl M. (1995), An Introduction to Analysis, Graduate Texts in Mathematics, vol. 154, Springer, p. 8, ISBN 9780387943695.

cite-note-33. ↑ citerefjech2003Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Springer Monographs in Mathematics, Springer, p. 34, ISBN 9783540440857.