Semi-ring of sets/Definition
From Maths
Definition
A collection of sets, F[Note 1] is called a semi-ring of sets if[1]:
- ∅∈F
- ∀S,T∈F[S∩T∈F]
- ∀S,T∈F∃(Si)mi=1⊆F pairwise disjoint[S−T=⋃⋅mi=1Si][Note 2] - this doesn't require S−T∈F note, it only requires that their be a finite collection of disjoint elements whose union is S−T.
Notes
- Jump up ↑ An F is a bit like an R with an unfinished loop and the foot at the right. "Semi Ring".
- Jump up ↑ Usually the finite sequence (Si)∞i=m⊆F being pairwise disjoint is implied by the ⋃⋅ however here I have been explicit. To be more explicit we could say:
- ∀S,T∈F∃(Si)mi=1⊆F[(∀i,j∈{1,…,m}⊂N[i≠j⟹Si∩Sj=∅])⏟the Si are pairwise disjointand⏞∧(S−T=⋃mi=1Si)]
- Caution:The statement: ∀S,T∈F∃(Si)mi=1⊆F[(∀i,j∈{1,…,m}⊂N[i≠j⟹Si∩Sj=∅])⟹(S−T=⋃mi=1Si)] is entirely different
- In this statement we are only declaring that a finite sequence exists, and if it is NOT pairwise disjoint, then we may or may not have S−T=⋃mi=1Si. We require that they be pairwise disjoint AND their union be the set difference of S and T.
- Caution:The statement: ∀S,T∈F∃(Si)mi=1⊆F[(∀i,j∈{1,…,m}⊂N[i≠j⟹Si∩Sj=∅])⟹(S−T=⋃mi=1Si)] is entirely different
- ∀S,T∈F∃(Si)mi=1⊆F[(∀i,j∈{1,…,m}⊂N[i≠j⟹Si∩Sj=∅])⏟the Si are pairwise disjointand⏞∧(S−T=⋃mi=1Si)]
