# 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:

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

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

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

### 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**.

As there are a number of conflicting definitions for the word **complex** in the context of group theory, it is highly recommended that the word *not* be used on $\mathsf{Pr} \infty \mathsf{fWiki}$ in this context.

## Examples

Let $G$ be a group.

### Example 1

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

Let:

\(\ds X\) | \(=\) | \(\ds \set {e, a^2}\) | ||||||||||||

\(\ds Y\) | \(=\) | \(\ds \set {e, a, a^3}\) |

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:

\(\ds X\) | \(=\) | \(\ds \set {e, a^2}\) | ||||||||||||

\(\ds Y\) | \(=\) | \(\ds \set {e, a, a^3}\) |

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.

### Example 4

Let $S$ be the initial segment of the natural numbers $\N_{<3}$:

- $\N_{<3} = \set {0, 1, 2}$

Let $\circ$ be the operation defined on $S$ by the Cayley table:

- $\begin {array} {c|cccc}

\circ & 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 2 \\ 1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ \end {array}$

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

Then every non-empty subset of $S$ which does not contain $0$ is invertible in $\struct {\powerset S, \circ_\PP}$.

### Subset Product with Empty Set

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

Let $A, B \in \powerset S$.

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

### Subsets of $\R$ under Multiplication

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

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

Let $T = \set {1, 2, 3}$.

Then the subset product $S T$ is:

- $ST = \set {-1, -2, -3, 2, 4, 6}$

## Also see

- Definition:Direct Image Mapping
- Definition:Minkowski Sum: for when $S$ is a vector space and the operation is vector addition

- Results about
**subset products**can be found**here**.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.9$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.1$: Subrings: Notation $3$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 27$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 41$: Multiplication of subsets of a group - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Definition $5.16$