Definition:Projection (Mapping Theory)/First Projection

Definition

Let $S$ and $T$ be sets.

Let $S \times T$ be the Cartesian product of $S$ and $T$.

The first projection on $S \times T$ is the mapping $\operatorname{pr}_1: S \times T \to S$ defined by:

$\forall \left({x, y}\right) \in S \times T: \operatorname{pr}_1 \left({x, y}\right) = x$

Also known as

This is sometimes referred to as the projection on the first co-ordinate.

Also see

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

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 6$: Ordered Pairs
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 8$: Functions
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 13$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$: Exercise $\text{R}$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 5$: Example $5.5$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.4$