In set theory, a projection is one of two closely related types of functions or operations, namely:
- A set-theoretic operation typified by the
th projection map, written
that takes an element
of the Cartesian product
to the value
[1] - A function that sends an element
to its equivalence class under a specified equivalence relation
[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
when
is understood, or written as
when it is necessary to make
explicit.
See also
References
- ^ 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.