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^th projection map, written $\mathrm{proj}_{j}\!$, that takes an element $\vec{x} = (x_1,\ \ldots,\ x_j,\ \ldots,\ x_k)$ of the Cartesian product $(X_1 \times \cdots \times X_j \times \cdots \times X_k)$ to the value $\mathrm{proj}_{j}(\vec{x}) = x_j$.

• A function that sends an element x to its equivalence class under a specified equivalence relation E. The result of the mapping is written as [x] when E is understood, or written as [x]_E when it is necessary to make E explicit.

See also

• Cartesian product
• Projection (relational algebra)
• Projection (mathematics)
• Relation