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 [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]
• A function that sends an element [math]\displaystyle{ x }[/math] to its equivalence class under a specified equivalence relation [math]\displaystyle{ E, }[/math] or, equivalently, a surjection from a set to another set. 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

• Cartesian product – Mathematical set formed from two given sets
• Projection (mathematics)
• Projection (measure theory)
• Projection (linear algebra) – Idempotent linear transformation from a vector space to itself
• Relation (mathematics) – Relationship between two sets, defined by a set of ordered pairs