Projection (set theory) explained

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,} that takes an element \vec{x} = (x _{1, ..., x} _{j, ..., x} _{k)} of the Cartesian product (X _{1} x … x X _{j} x … x X _{k)} to the value 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.

Notes and References

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