Projection (set theory)

The ( canonical ) projection , projection mapping , coordinate mapping or evaluation mapping is a mapping in mathematics that maps a tuple onto one of the components of the tuple. More generally, a projection is a mapping from the Cartesian product of a family of sets onto the Cartesian product of a subfamily of these sets that selects elements with certain indices . Assuming the axiom of choice , a projection of any family of non-empty sets is always surjective . Projections are used, among other things, in set theory , in topology , in measure theory or as operators in relational databases .

definition

If is a family of sets , where is an arbitrary index set , then denotes the Cartesian product of these sets. If now is a subset of , then the projection onto this subset is the mapping ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle I}$ ${\displaystyle \textstyle X_{I}=\prod _{i\in I}X_{i}}$ ${\displaystyle J\subseteq I}$ ${\displaystyle I}$ ${\displaystyle \pi _{J}}$

${\displaystyle \pi _{J}\colon X_{I}\to X_{J},\quad (x_{i})_{i\in I}\mapsto (x_{j})_{j\in J}}$.

The projection therefore selects those elements from a family of elements whose indices are contained in the set . In the case of a one-element set , the projection is also simply noted through . ${\displaystyle \pi _{J}}$ ${\displaystyle (x_{i}\in X_{i})_{i\in I}}$ ${\displaystyle J}$ ${\displaystyle J=\{j\}}$ ${\displaystyle \pi _{\{j\}}}$ ${\displaystyle \pi _{j}}$

Examples

Orderly couples

If the index set consists of exactly two elements, then the Cartesian product is the set of the ordered pairs of elements of the two sets and . The projections ${\displaystyle I=\{1,2\}}$ ${\displaystyle X_{I}=X_{1}\times X_{2}}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$

${\displaystyle \pi _{1}\colon X_{1}\times X_{2}\to X_{1},\quad (x_{1},x_{2})\mapsto x_{1}}$

and

${\displaystyle \pi _{2}\colon X_{1}\times X_{2}\to X_{2},\quad (x_{1},x_{2})\mapsto x_{2}}$

then map a pair onto its first or second component. For example, are the Cartesian coordinates of a point in the Euclidean plane , the projections yield and respectively - and the coordinate of the point. These projections are to be formally differentiated from ( orthogonal ) projections on the two coordinate axes that images with or represent. ${\displaystyle (x_{1},x_{2})}$ ${\displaystyle (x,y)\in \mathbb {R} ^{2}}$ ${\displaystyle \pi _{1}}$ ${\displaystyle \pi _{2}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{2}}$ ${\displaystyle (x,y)\mapsto (x,0)}$ ${\displaystyle (x,y)\mapsto (0,y)}$

Tuple

If the index set consists of elements, then the Cartesian product is the set of all - tuples in which the -th component is an element . The projection is then the image ${\displaystyle n}$ ${\displaystyle I=\{1,\ldots ,n\}}$ ${\displaystyle X_{I}=X_{1}\times \ldots \times X_{n}}$ ${\displaystyle n}$ ${\displaystyle i}$ ${\displaystyle x_{i}\in X_{i}}$ ${\displaystyle \pi _{j}}$

${\displaystyle \pi _{j}\colon X_{1}\times \ldots \times X_{n}\to X_{j},\quad (x_{1},\ldots ,x_{n})\mapsto x_{j}}$,

which a tuple maps to its -th component. Each tuple thus has the representation . ${\displaystyle j}$ ${\displaystyle T\in X_{I}}$ ${\displaystyle T=(\pi _{1}(T),\ldots ,\pi _{n}(T))}$

Functions

If the sets are all equal to a set , then the Cartesian product is the set of all functions . The projection is then the image ${\displaystyle X_{i}}$ ${\displaystyle X}$ ${\displaystyle X_{I}=X^{I}}$ ${\displaystyle f\colon I\to X}$ ${\displaystyle \pi _{j}}$

${\displaystyle \pi _{j}\colon X^{I}\to X,\quad f\mapsto \pi _{j}(f)=f(j)}$,

which maps a function to its function value for the argument . This mapping is therefore also referred to as an evaluation mapping. ${\displaystyle j}$

properties

Surjectivity

If the index set is finite and the sets are not empty , then a projection mapping is always surjective , that is ${\displaystyle I}$ ${\displaystyle X_{i}}$

${\displaystyle \pi _{J}(X_{I})=X_{J}}$.

To ensure that the Cartesian product of any family of non-empty sets is also non-empty, however, the axiom of choice is required. In fact, the above statement is even equivalent to the axiom of choice. Assuming the axiom of choice, a projection mapping is then always surjective for any family of non-empty sets.

Archetype

If it is a real subset of the index set and is a subset of the target set of a projection , then the archetype of has the representation ${\displaystyle J\subset I}$ ${\displaystyle I}$ ${\displaystyle \textstyle W\subseteq X_{J}}$ ${\displaystyle \pi _{J}}$ ${\displaystyle W}$

${\displaystyle \pi _{J}^{-1}(W)=W\times X_{I\setminus J}=\{(x_{i})_{i\in I}\in X_{I}\mid (x_{j})_{j\in J}\in W\}}$.

The quantities are also referred to as cylinder quantities . ${\displaystyle \pi _{J}^{-1}(W)}$

use

topology

If for topological spaces , then the product topology is based on the coarsest topology (the topology with the fewest open sets ) with respect to which all projections are continuous . The cylinder sets of the form , where is an open subset of , form a sub-basis for the product space . The product space can also be characterized by the following universal property of a categorical product : if it is a topological space and the mapping for each is continuous, then there is exactly one continuous function , so that ${\displaystyle X_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle X_{I}}$ ${\displaystyle \pi _{j}}$ ${\displaystyle \pi _{j}^{-1}(U_{j})}$ ${\displaystyle U_{j}}$ ${\displaystyle X_{j}}$ ${\displaystyle X_{I}}$ ${\displaystyle Y}$ ${\displaystyle f_{j}\colon Y\to X_{j}}$ ${\displaystyle j\in I}$ ${\displaystyle f\colon Y\to X_{I}}$

${\displaystyle \pi _{j}\circ f=f_{j}}$

applies to all . Conversely, a given function is continuous if and only if all projections are continuous. In addition to continuity, the projections are open mappings , i.e. each open subspace of the product space remains open when it is projected onto a set . However, the converse does not apply: if a subspace of the product space, whose projections are all open, then even it does not have to be open. In general, the projections are also not self-contained images. ${\displaystyle j\in I}$ ${\displaystyle f\colon Y\to X_{I}}$ ${\displaystyle \pi _{j}\circ f}$ ${\displaystyle \pi _{j}\colon X_{I}\to X_{j}}$ ${\displaystyle W\subset X_{I}}$ ${\displaystyle X_{I}}$ ${\displaystyle X_{j}}$ ${\displaystyle W\subset X_{I}}$ ${\displaystyle \pi _{j}\colon W\to X_{j}}$ ${\displaystyle W}$ ${\displaystyle X_{I}}$ ${\displaystyle \pi _{j}\colon X_{I}\to X_{j}}$

Measure theory

If for measurement spaces , then the product is σ-algebra ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle i\in I}$

${\displaystyle \bigotimes _{i\in I}{\mathcal {A}}_{i}=\sigma \left(\left\{\pi _{j}^{-1}(A_{j})\mid A_{j}\in {\mathcal {A}}_{j},j\in I\right\}\right)=\sigma \left(\bigcup _{j\in I}\pi _{j}^{-1}({\mathcal {A}}_{j})\right)}$

the smallest σ-algebra on the Cartesian product , so that all projections onto the individual quantities are measurable . The product σ-algebra is also generated by the system of all cylinder sets with a finite index set . In the measure theory and stochastic product σ-algebra form the basis for product dimensions and product probability spaces . ${\displaystyle \Omega _{I}}$ ${\displaystyle \Omega _{i}}$ ${\displaystyle J}$

Computer science

Projections are also used as operators in relational databases . If this is a relation and a subset of the attribute set , then the result of the projection is ${\displaystyle R}$ ${\displaystyle \{A_{1},\ldots ,A_{k}\}}$

${\displaystyle \Pi _{A_{1},\ldots ,A_{k}}(R)=\{T[A_{1},\ldots ,A_{n}]\mid T\in R\}}$

a new relation that only contains the attributes from the specified attribute list. Duplicate entries are deleted in the result relation.

See also

• Restriction
• Inclusion illustration
• Quotient mapping
• Fiber bundle

literature

• Gerd Fischer : Linear Algebra: an introduction for first-year students . Springer, 2008, ISBN 3-8348-9574-1 .
• Paul Halmos : Naive set theory . Springer, 1960, ISBN 0-387-90092-6 .
• Norbert Kusolitsch: Measure and probability theory . Springer, 2014, ISBN 978-3-642-45387-8 .
• Jochen Wengenroth: Probability Theory . de Gruyter, 2008, ISBN 978-3-11-020359-2 .
• Stephen Willard: General Topology . Courier Dover Publications, 2012, ISBN 978-0-486-13178-8 .

Individual evidence

1. ^ ^{A b} Paul Halmos: Naive set theory . Springer, 1960, p. 36 .
2. ↑ Gerd Fischer: Lineare Algebra: an introduction for first-year students . Springer, 2008, p. 38 .
3. ↑ ^{a b} Jochen Wengenroth: Probability Theory . de Gruyter, 2008, p. 14 .
4. ↑ Stephen Willard: General Topology . Courier Dover Publications, 2012, pp. 52 .
5. ↑ Norbert Kusolitsch: Measure and probability theory . Springer, 2014, p. 6 .