# Definition:Subset Product

## Definition

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

We can define an operation on the power set $\powerset S$ as follows:

$\forall A, B \in \powerset S: A \circ_\mathcal P B = \set {a \circ b: a \in A, b \in B}$

This is called the operation induced on $\powerset S$ by $\circ$, and $A \circ_\mathcal P B$ is called the subset product of $A$ and $B$.

It is usual to write $A \circ B$ for $A \circ_\mathcal P B$.

If $A = \O$ or $B = \O$, then $A \circ B = \O$.

### Subset Product with Singleton

When one of the subsets in a subset product is a singleton, we can (and often do) dispose of the set braces. Thus:

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

Then:

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

## Also defined as

When the operation $\circ$ is in fact addition ($+$), the induced operation $+$ is then also called setwise addition.

When used, it is best to state explicitly that $+$ means setwise addition.

This is because some sources use $A + B$ also to denote set union and disjoint union.

## Also known as

Also known as a complex.

## Examples

Let $G$ be a group.

### Example 1

Let $a \in G$ be an element of $G$.

Let:

 $\displaystyle X$ $=$ $\displaystyle \set {e, a^2}$ $\displaystyle Y$ $=$ $\displaystyle \set {e, a, a^3}$

Let $\order a = 4$.

Then:

$\card {X Y} = 4$

where $\card {\, \cdot \,}$ denotes cardinality.

### Example 2

Let $a \in G$ be an element of $G$.

Let:

 $\displaystyle X$ $=$ $\displaystyle \set {e, a^2}$ $\displaystyle Y$ $=$ $\displaystyle \set {e, a, a^3}$

Let $\order a = 6$.

Then:

$\card {X Y} = 5$

where $\card {\, \cdot \,}$ denotes cardinality.

### Example 3

Let the order of $G$ be $n \in \Z_{>0}$.

Let $X \subseteq G$ be a subset of $G$.

Let $\card X > \dfrac n 2$.

Then:

$X X = G$

where $X X$ denotes subset product.

## Also see

• Results about Subset Products can be found here.