Projection (set theory)

# Projection (set theory)

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In set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, a projection is one of two closely related types of function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s or operations, namely:
• A set-theoretic
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

operation typified by the jth projection map, written , that takes an element of the cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

to the value .

• A function that sends an element x to its equivalence class under a specified equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

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.