Definition:Image of Mapping/Definition 2

Definition

The image of a mapping $f: S \to T$ is the set:

$\Img f = f \sqbrk S$

where $f \sqbrk S$ is the image of $S$ under $f$.

Also denoted as

The notation $\Img f$ to denote the image of a mapping $f$ is specific to $\mathsf{Pr} \infty \mathsf{fWiki}$.

The usual notation is $\map {\mathrm {Im} } f$ or a variant, but this is too easily confused with $\map \Im z$, the imaginary part of a complex number.

Hence the non-standard usage $\Img f$.

Some sources use $f \sqbrk S$, where $S$ is the domain of $f$.

Others just use $\map f S$, but that notation is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$ so as not to confuse it with the notation for the image of an element.

Also see

• Equivalence of Definitions of Image of Mapping

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 11$
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 21.1$: The image of a subset of the domain; surjections
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions