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 $\mathrm {proj} _{j},$ {\displaystyle \mathrm {proj} _{j},} that takes an element ${\vec {x}}=(x_{1},\ \dots ,\ x_{j},\ \dots ,\ x_{k})$ {\displaystyle {\vec {x}}=(x_{1},\ \dots ,\ x_{j},\ \dots ,\ x_{k})} of the Cartesian product $(X_{1}\times \cdots \times X_{j}\times \cdots \times X_{k})$ {\displaystyle (X_{1}\times \cdots \times X_{j}\times \cdots \times X_{k})} to the value $\mathrm {proj} _{j}({\vec {x}})=x_{j}.$ {\displaystyle \mathrm {proj} _{j}({\vec {x}})=x_{j}.}^[1]
A function that sends an element $x$ {\displaystyle x} to its equivalence class under a specified equivalence relation $E,$ {\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 $[x]$ {\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.