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 [math]\displaystyle{ j }[/math]th projection map, written [math]\displaystyle{ \mathrm{proj}_j, }[/math] that takes an element [math]\displaystyle{ \vec{x} = (x_1,\ \dots,\ x_j,\ \dots,\ x_k) }[/math] of the Cartesian product [math]\displaystyle{ (X_1 \times \cdots \times X_j \times \cdots \times X_k) }[/math] to the value [math]\displaystyle{ \mathrm{proj}_j(\vec{x}) = x_j. }[/math][1]
  • A function that sends an element [math]\displaystyle{ x }[/math] to its equivalence class under a specified equivalence relation [math]\displaystyle{ E, }[/math][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 [math]\displaystyle{ [x] }[/math] when [math]\displaystyle{ E }[/math] is understood, or written as [math]\displaystyle{ [x]_E }[/math] when it is necessary to make [math]\displaystyle{ E }[/math] explicit.

See also


  1. Halmos, P. R. (1960), Naive Set Theory, Undergraduate Texts in Mathematics, Springer, p. 32, ISBN 9780387900926, .
  2. Brown, Arlen; Pearcy, Carl M. (1995), An Introduction to Analysis, Graduate Texts in Mathematics, 154, Springer, p. 8, ISBN 9780387943695, .
  3. Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Springer Monographs in Mathematics, Springer, p. 34, ISBN 9783540440857, .