# 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 jth 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.