This theory is all set(s)

Set theory is, along with geometry, arithmetic and algebra, one of the major foundations of mathematics.  It is described by multi-valued (possibly infinitely-valued) Boolean algebras.

A set is a collection of things (objects, abstractions, other sets, etc.).  The things that make up a set are called members or elements of that set.  The set which consists of all things that may be members of the sets being considered is called the universe of discourse.

Membership in a set is described by a Boolean algebra in which

• Members of the set belong to the set of values of the Boolean algebra.
• The intersection of two sets, a set whose members are members of both sets, is the conjunction operation of the Boolean algebra.
• The union of two sets, a set whose members are members of either (or both) sets, is the disjunction operation of the Boolean algebra.
• The complement of a set, which consists of of those elements in the universe of discourse that are not members of the set, is the complement operation of the Boolean algebra.
• The empty set, the set which does not contain any members, is the zero of the Boolean algebra.
• The universe of discourse is the unit of the Boolean algebra.

The subset relation, which holds if every member of the first set is a member of the second set, is the (partial) order of the Boolean algebra.  If each set has one or more members that are not members of the other set, the subset relation does not hold in either order.

Propositional algebra:
                             1
                             |
                             0

The table of the order relations for this graph is

    Right | 0  1 
Left      |------
  0       | =  <
  1       | >  =  

where the lower end of a path is 'less than' the upper end of the path.

Set theory:
                          {a, b, c}
                         /    |   \
                        /     |    \
                   {a, b}  {a, c}  {b, c}
                     |  \  /   \  /  |
                     |   \/     \/   |
                     |   /\     /\   |
                     |  /  \   /  \  |
                     {a}    {b}    {c}
                        \    |    /
                         \   |   /
                          \  |  /
                            { }

The table of the order relations for this graph is
      Right { } {a} {b} {c} {a, b} {a, c} {b, c} {a, b, c}
Left      |----------------------------------------------- 
      { } |  =   <   <   <    <      <      <        <
      {a} |  >   =   X   X    <      <      X        <
      {b} |  >   X   =   X    <      X      <        <
      {c} |  >   X   X   =    X      <      <        <
   (a, b} |  >   >   >   X    =      X      X        <
   (a, c} |  >   >   X   >    X      =      X        <
   {b, c} |  >   X   >   >    X      X      =        <
{a, b, c} |  >   >   >   >    >      >      >        =


where < means 'left' is a subset of 'right', > means 'right' is a subset of 'left', = means 'left' and 'right' are the same, and 'X' means none of these relations apply.