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 j^th projection map, written $\mathrm{proj}_{j}\!$ , that takes an element $\vec{x} = (x_1,\ \ldots,\ x_j,\ \ldots,\ x_k)$ of the cartesian product $(X_1 \times \cdots \times X_j \times \cdots \times X_k)$ to the value $\mathrm{proj}_{j}(\vec{x}) = x_j$ .

A function that sends an element x to its equivalence class under a specified equivalence relation E. The result of the mapping is written as [x] when E is understood, or written as [x]_E when it is necessary to make E explicit.

This page was last modified 01:11, 20 January 2008 by Wikipedia user Lambiam. Based on work by Wikipedia user(s) Michael Hardy, Charles Matthews, Oleg Alexandrov and Jon Awbrey and Anonymous user(s) of Wikipedia.
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