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 {\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.