set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, 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_j,

that takes an element

\vec = (x_1,\ \dots,\ x_j,\ \dots,\ x_k)

of the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

(X_1 \times \cdots \times X_j \times \cdots \times X_k)

to the value

\mathrm_j(\vec) = x_j.

* A function that sends an element

x

to its

equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

under a specified

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

E,

or, equivalently, a surjection from a set to another set.. 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

/math> when E is understood, or written as E when it is necessary to make E explicit.

References

Basic concepts in set theory {{settheory-stub

See also

References