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}.}
[1]
- A function that sends an element
x
{\displaystyle x}
to its equivalence class under a specified equivalence relation E , {\displaystyle 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 ] {\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}
See also
- 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
References
- Halmos, P. R. (1960), Naive Set Theory, Undergraduate Texts in Mathematics, Springer, p. 32, ISBN 9780387900926
{{citation}}: ISBN / Date incompatibility (help). - 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.