# Definition:Support of Mapping to Algebraic Structure

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## Contents

## 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:

- $\operatorname{supp} \left({f}\right) := \left\{{x \in S: f \left({x}\right) \ne 0}\right\}$

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

### General Algebraic Structure

Let $(A, *)$ 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:

- $\operatorname{supp}(f) = \{s \in S : f(s) \neq e\}$

## Sequence

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