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