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 {\displaystyle j} th projection map, written p r o j j , {\displaystyle \mathrm {proj} _{j},} that takes an element x → = ( x 1 ,   … ,   x j ,   … ,   x k ) {\displaystyle {\vec {x}}=(x_{1},\ \dots ,\ x_{j},\ \dots ,\ x_{k})} of the Cartesian product ( X 1 × ⋯ × X j × ⋯ × X k ) {\displaystyle (X_{1}\times \cdots \times X_{j}\times \cdots \times X_{k})} to the value p r o j j ( x → ) = x j . {\displaystyle \mathrm {proj} _{j}({\vec {x}})=x_{j}.}
• A function that sends an element x {\displaystyle x} to its equivalence class under a specified equivalence relation E , {\displaystyle E,} 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 [ x ] {\displaystyle [x]} when E {\displaystyle E} is understood, or written as [ x ] E {\displaystyle [x]_{E}} when it is necessary to make E {\displaystyle E} explicit.