Everything on this page depends on [ logic_30_Sets.html ] for syntax, semantics and properties.

The Power Set of a Type

For Type T.

1. For S:@T, @S::= the set of subsets of a set S,
2. For S, @S::@@T= @^ S, the set of subsets of S, also known as the power set.
3. For S, SetOf(S)::= @S, and form used with audiences unused to MATHS and/or set theory.
   
4. (above)|-{1,3} and {2} in SetOf({1,2,3}).

   
   

5. (above)|-@S={B:@T || B==>S}.
   
6. (above)|-|@^S|=|@|^|S|=2^|S|,

   The Set of families of sets

   A family is a collection of subsets of a given set - Given a set S of elements {x,y,...} then @S is the set of all subsets of S, @S = {{},{x}, {x,y},{y},...}. @@S is short for @(@S) - the set of subsets of subsets of S. Thus @@S={{}, {{}}, {{},{x}}, {{},{y}}, {{x},{y}},...}. An element of @@S is a called a family of subsets of S. It turns out that many important ideas can be best modelled as particular kinds of families - in particular, abstract families of languages, families of open and closed sets, partitions and covers all turn out to be important.

   
   

7. |- (union): For β:@@T, |(β)={x:T || for some B:β (x in B)},
   
8. |- (intersection): For β:@@T, &(β)={x:T || for all B:β(x in B)}.

   Special Types of Families

   Covers

   Families that have every element in them at least once, but may overlap...
   1. For S:Sets, covers(S)::={α:@@S || |(α)=S},
      
   2. (covers)|-{S} in covers(S),
      
   3. (covers)|-for A:@S {A,S~A} in covers(S),
      
   4. (covers)|-for α, β:covers(S)(α | β in covers(S)),
      
   5. (covers)|-for α:covers(S), β:@@S, if α==>β then β in covers(S)).

      Simplicial Complexes

      If S is a finite set and C:@S, then we can call each element of C a simplex. When we do this we imply that the elements of C are classified by the number of elements in them:
   6. For c:C, dim(c)::=order(c) ::= |C|-1, for p:0..|S|-1, p.simplex={c:C || dim(c)=p}.

      A face f of a simplex c in C is any simplex which is subset of c.

   7. For p,c, p.face(c)::=@c and p.simplex. For p,c, faces(c)=|[p](p.face(c)).

      There are several definitions of a simplicial complex. According to Ron Atkin[Casti 92, Atkin 74]:

   8. (atkin): C in simplicial_complex iff S==>C and for all c:C(faces(c) in C).

      The dimension of a simplicial complex C, dim(C) ::=max(dim(C)). If dim(C)=n then S can be embedded in a Euclidean Space of 2n+1 dimensions so that

   9. (1) each simplex of dimension p is a convex polytope of dimension p
   10. (2) the convex polytopes overlap iff the simplices do.

       Given a family C of subsets of a finite set S then we can construct an Atkin-complex by adding all subsets of elements of C to C:

   11. complex(C)::={x:@c || for some c:C }.

       Another definition [James & James 68] is C is a simplicial_complex iff for all c,c':C, either c&c'={} or c&c' in faces(c) and faces(c').

       Simplicial Complexes are generated by relations between finite sets [ logic_42_Properties_of_Relation.html ] for example.

       Mosaics

       A mosaic is a collection of mutually disjoint subsets of a set that may or may not cover the set. Botting 1970

   12. For S:Sets, mosaics(S)::={μ:@@S || for all A,B:μ, if A<>B then A&B={}}.
   13. For S:Sets, μ:mosaics(S), free(μ)::=S~ |(μ).
       
   14. (above)|-for S, ( {S},{{}} )in mosaics(S) ).
       
   15. (above)|-for A:@S, {A,S~A} in mosaics(S) and free({A,S~A} )={}.
       
   16. (above)|-for μ:mosaics(S), ν:@μ (ν in mosaics(S)).

       Partitions

       Mutually disjoint covers.
   17. For S:Sets, partitions(S)::=covers(S) & mosaics(S).
       
   18. (above)|-For μ:mosaics(S) ( (μ | free(μ) ) in partitions(S) ),
   19. For π:@S, S>==π::=for all a:S, one b:π(a in π),
       
   20. (above)|-For all π: partition(S), S>==π,
       
   21. (above)|-For all A,B:@S, {A&B,A~B} in partition(S).
   22. For S,π, s:S, π(s)=the [A:π]( s in A ).
       
   23. (above)|-For S, π, rel[a,b](π(a)=π(b)) in eguivalence_relation.

       [ Equivalence Relations in logic_41_Maps ]

       Open and closed families

       .Used_in topology(continuity, limits,...). [ math_91_Topology.html ]

       Useful whenever the idea of limit is needed to define what happens to iterative and recursive processes.

   24. For S:Sets, open_family(S):@@S::= { open:@S || (open,&,{}) in monoid and for G:@open(|(G) in open) and S in open },
       
   25. |-for all open:open_family(S), G:finite(open)(&(G) in open).
   26. For S:Sets, closed_family(S):@@S::= { closed:@S || (closed,|,S) in monoid and for all G:@closed( &(G) in closed) and {} in closed },
       
   27. |-for closed:closed_family(S), G:finite(closed) (|(G) in closed).

   28. (Duality): for α:@@S, β={ S~A || A:α} ( α in closed_family(S) iff β in open_family(S))

       Filters

       Bourbaki's Abstract Model of x tends to x0. A filter is closed under finite intersections and supersetting

       .Used_in Math.Topology. [ math_91_Topology.html ]

       1. For S:Sets, filters(S)::={f:@(@S-{}) || for all B:@S(if for some A:f(A==>B) then B in f) and for all A,B:f ( A & B in f) },

          
          

       2. (above)|-for all G:finite(f) ( &(G) in f).

          Filter Bases

          1. For S:Sets, filter_bases(S)::={f:@(@S-{}) || for all A,B:f, some C:f(C==>A&B) },
             
          2. (above)|-for all f:filter_bases(S), A,B:f (A & B).
             
          3. (above)|-for all f:filter_bases(S) ({A:@S || for some B:f(B==>A)} in filter(S) ).

             Every set in a filterbase has its own filterbase,...
             

          4. (above)|-for all f:filter_bases(S), F:f, {A&F || A:f} in filter_bases(F).

             Ultra-Filters

          5. For S:Sets, ultrafilters(S)::={f:filters(S) || for all A:@S(A in f or S~A in f) }.

          . . . . . . . . . ( end of section Filters Bases) <<Contents | End>>

       . . . . . . . . . ( end of section Filters) <<Contents | End>>

       Lattices of Sets

       An abstract lattice has two operations that are idempotent and absorbative.

       A set of sets is a lattice with respect to union and intersection iff every union and intersection of every pair of sets in the family is also in the family.

       This is the same as requiring that all finite intersections and unions are members of the family.

       Complete Lattices of Sets

       A collection of sets is a complete lattice is all unions and intersections of all sets of sets in the family are also in the family ... even with infinite unions and interesections.
       1. generate(α)::=the smallest lattice of sets containing all elements of α.

          Bracketing sets

          A pair of sets in a complete lattice L form a bracket if the first is a subset (or is equal to) the other:
       2. Bracket(L)::@(L><L)={ (B,T):L><L. B ==> T }.

          Each Bracket (B,T) determines a sublattice { U:L. B==>U==>T } of L.

          Rough sets

          A rough set that approximates set Target in lattice L is the bracket (Lower,Upper) where
       3. Lower:= &{ X:L . Target ==> X},
       4. Upper:= |{ X:L . X ==>Target }.

          rough_set[L](T) ::=(&{ X:L . Target ==> X}, |{ X:L . X ==>Target }).

          
          

       5. |-rough_set[L](T) in Bracket(L).

          Lattice based Partition sets

          Given a map f from set X to Y, lattice(f) ::= generate(X/f).

          Given maps a:X>->attributes (assumed normalized), b>->decisions,

       6. rough_sets(c,L)::= { rough_set(L)(c) || c: X/b }.

       . . . . . . . . . ( end of section Lattices of Sets) <<Contents | End>>

   . . . . . . . . . ( end of section Special types of Families) <<Contents | End>>

   Refinement of families

9. For S:Sets, α:@@S, refinement(α )::={ β in @@S || for all B in β, some A in α (B==>A)}.

10. For S:Sets, α,β:@@S, (β finer α ) ::=for all B in β, some A in α (B==>A).

. . . . . . . . . ( end of section Families of Sets/Hypergraphs) <<Contents | End>>

Notes on MATHS Notation

Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle = Net{radius:Positive Real, center:Point}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

Notes on the Underlying Logic of MATHS

For a more rigorous description of the standard notations see