Definition:Projection (Mapping Theory)/Family of Sets

Definition

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 $\ds \prod_{i \mathop \in I} S_i$.

Also known as

This is sometimes referred to as the projection on the $j$th co-ordinate.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 10$: Arbitrary Products