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 $\mathrm{proj}_{j}(\vec{x}) = x_j$.^[1]
• A function that sends an element x to its equivalence class under a specified equivalence relation 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] 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

References

1. Halmos, P. R. (1960), Naive Set Theory, Undergraduate Texts in Mathematics, Springer, p. 32, ISBN 9780387900926.
2. Brown, Arlen; Pearcy, Carl M. (1995), An Introduction to Analysis, Graduate Texts in Mathematics, 154, Springer, p. 8, ISBN 9780387943695.
3. Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Springer Monographs in Mathematics, Springer, p. 34, ISBN 9783540440857.