Definition:Projection (Mapping Theory)

This page is about projection in the context of mapping theory. 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

• Projection is Surjection
• Projection is Epimorphism

• The left operation and right operation for the same concept in the context of abstract algebra.

• Definition:Canonical Injection

• 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

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• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.4$