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 $proj\_\{j\}\backslash !$, that takes an element $\backslash vec\{x\}\; =\; (x\_1,\backslash \; \backslash ldots,\backslash \; x\_j,\backslash \; \backslash ldots,\backslash \; x\_k)$ of the cartesian product $(X\_1\; \backslash times\; \backslash cdots\; \backslash times\; X\_j\; \backslash times\; \backslash cdots\; \backslash times\; X\_k)$ to the value $proj\_\{j\}(\backslash vec\{x\})\; =\; x\_j$.

• 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 when E is understood, or written as [x_E when it is necessary to make E explicit.

See also

• Relation
• Tacit extension
• cartesian product

Set theory