Definition:Subset Product/Singleton
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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:
- $ST = \set {-3, 6}$
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
- 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