# Definition:Image (Set Theory)/Mapping/Element/Also known as

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## Definition

The image of an element $s$ under a mapping $f$ is also called the **functional value**, or **value**, of $f$ at $s$.

The terminology:

**$f$ maps $s$ to $\map f s$****$f$ assigns the value $\map f s$ to $s$****$f$ carries $s$ into $\map f s$**

can be found.

The modifier **by $f$** can also be used for **under $f$**.

Thus, for example, the **image of $s$ by $f$** means the same as the **image of $s$ under $f$**.

In the context of computability theory, the following terms are frequently found:

If $\tuple {x, y} \in f$, then $y$ is often called the **output** of $f$ for **input** $x$, or simply, the **output of $f$ at $x$**.

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 4$. Relations; functional relations; mappings:*Remark $1$* - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 20$: Introduction: Remarks $\text{(g)}$