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.