Subset Product/Examples
Examples of Subset Product
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:
- $ST = \set {-1, -2, -3, 2, 4, 6}$
Congruence Modulo $m$ in $\N_{<m}$
Let $m \in \Z_{>0}$ be a (strictly) positive integer.
Let $\N_{<m}$ denote the initial segment of the natural numbers $\N$:
- $\N_{<m} = \set {0, 1, \ldots, m - 1}$
Let $\RR_m$ denote the equivalence relation:
- $\forall x, y \in \Z: x \mathrel {\RR_m} y \iff \exists k \in \Z: x = y + k m$
For each $a \in \N_{<m}$, let $\eqclass a m$ be the equivalence class of $a \in \N_{<m}$ under $\RR_m$:
- $\eqclass a m := \set {a + z m: z \in \Z}$
Let $\Z_m$ be the set defined as:
- $\Z_m := \set {\eqclass a m: a \in \N_{<m} }$
Let $+_\PP$ denote the operation induced on $\powerset \Z$ by integer addition.
Then the algebraic structure $\struct {\Z_m, +_\PP}$ is closed in the sense:
- $\forall \eqclass a m, \eqclass b m \in \Z_m: \eqclass a m +_\PP \eqclass b m \in \Z_m$