# Definition:Subset Product/Singleton

## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $A \subseteq S$ be a subset of $S$.

Then:

$(1): \quad a \circ S := \set a \circ S$
$(2): \quad S \circ a := S \circ \set a$

where $\set a \circ S$ and $S \circ \set a$ denote the subset product of $\set a$ with $S$.

That is:

$a \circ S = \set {a \circ s: s \in S}$
$S \circ a = \set {s \circ a: s \in S}$

## Examples

### Subsets of $\R$ under Multiplication

Let $\struct {\R, \times}$ be the multiplicative group of (non-zero) real numbers.

Let $S = \set 3$.

Let $T = \set {-1, 2}$.

Then the subset product $S T$ is:

$ST = \set {-3, 6}$