Generating set
In
mathematics,
subset S of
algebraic structure G is
generating setG (or
G"generated" by
S) ifsmallest subset
G that includes
Sis closed underalgebraic operations on
GG itself. For example, if
G is
groupitself issmallest subgroup
G containing
S, then
S isgenerating set
G.
Examples
- The additive groupintegers has 1 asgenerating set. The element 2notgenerating set, asodd numbers will be missing. The two-elements subset {3, 5} isgenerating set.
- In linear algebra, S isgenerating set or spanning set ofvector space V if V islinear spanS.
- Continuous functions oninterval. Polynomials aregenerating set, because closure under limits formsentire space. (we needconceptclosure undergiven topology here)
