Definition:Projection (Mapping Theory)
This page is about Projection in the context of Finite Cartesian Product. For other uses, see Projection.
Definition
Let $S_1, S_2, \ldots, S_j, \ldots, S_n$ be sets.
Let $\ds \prod_{i \mathop = 1}^n S_i$ be the Cartesian product of $S_1, S_2, \ldots, S_n$.
For each $j \in \set {1, 2, \ldots, n}$, the $j$th projection on $\ds S = \prod_{i \mathop = 1}^n S_i$ is the mapping $\pr_j: S \to S_j$ defined by:
- $\map {\pr_j} {s_1, s_2, \ldots, s_j, \ldots, s_n} = s_j$
for all $\tuple {s_1, s_2, \ldots, s_n} \in S$.
Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets.
Let $\ds \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.
For each $j \in I$, the $j$th projection on $\ds S = \prod_{i \mathop \in I} S_i$ is the mapping $\pr_j: S \to S_j$ defined by:
- $\map {\pr_j} {\family {s_i}_{i \mathop \in I} } = s_j$
where $\family {s_i}_{i \mathop \in I}$ is an arbitrary element of $\ds \prod_{i \mathop \in I} S_i$.
Projection from Product of Two Sets
The definition is most usually seen in the context of the Cartesian product of two sets, as follows.
Let $S$ and $T$ be sets.
Let $S \times T$ be the Cartesian product of $S$ and $T$.
First Projection
The first projection on $S \times T$ is the mapping $\pr_1: S \times T \to S$ defined by:
- $\forall \tuple {x, y} \in S \times T: \map {\pr_1} {x, y} = x$
Second Projection
The second projection on $S \times T$ is the mapping $\pr_2: S \times T \to T$ defined by:
- $\forall \tuple {x, y} \in S \times T: \map {\pr_2} {x, y} = y$
Also denoted as
It is common to denote projections with the Greek letter $\pi$ (pi) in place of $\pr$.
Some sources, in particular those which approach the subject from the context of physics, use a superscript: $\pr^j$.
Some sources use the notation $p_j$ for $\pr$.
Also known as
This is sometimes referred to as the projection on the $j$th co-ordinate, and hence generically as a coordinate projection.
Some sources use a $0$-based system to number the elements of a Cartesian product.
For a given ordered $n$-tuple $x = \tuple {a_0, a_1, \ldots, a_{n - 1} }$, the notation $\paren x_j$ is also seen.
Hence:
- $\paren x_j = a_j$
which is interpreted to mean the same as:
- $\map {\pr_j} {a_0, a_1, \ldots, a_{n - 1} } = a_{j - 1}$
On $\mathsf{Pr} \infty \mathsf{fWiki}$, to avoid all such confusion, the notation $\map {\pr_j} {a_1, a_2, \ldots, a_n} = a_j$ is to be used throughout.
Also see
- The left operation and right operation for the same concept in the context of abstract algebra.
- Results about projections can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 3.1$: Direct sums
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.4$: Sets of Sets
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.4$