# Definition:Support of Mapping to Algebraic Structure

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## Definition

### Real-Valued Function on an Abstract Set

Let $S$ be a set.

Let $f: S \to \R$ be a real-valued function.

The **support of $f$** is the set of elements $x$ of $S$ whose values under $f$ are non-zero.

That is:

- $\map \supp f := \set {x \in S: \map f x \ne 0}$

### General Real-Valued Function in $\R^n$

### General Algebraic Structure

Let $\struct {A, *}$ be an algebraic structure with an identity element $e$.

Let $S$ be a set.

Let $f: S \to A$ be a mapping.

The **support of $f$** is the set:

- $\map \supp f = \set {s \in S : \map f s \ne e}$

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## Sequence

Note that by definition, a sequence is a mapping, so that the definition of support applies in particular to sequences.

## Also denoted as

The **support of $f$** can also be seen denoted as $\map {\mathrm {Supp} } f$.