To say that sets A and B are equal means that they have the same elements, i.e. A = B means ‘meA implies and is implied by meB’.
A set R is a subset of (or contained in) a set S if every element of R is an element of S. We write this relation RcS. From the definition it follows that any set is a subset of itself, that is ScS, that is the relation c is reflexive. If RcS and ScR then it follows that R=S, that is the relation c is antisymmetric. If RcS and ScT then it follows that RcT, that is the relation c is transitive. Since the relation of being a subset is reflexive, antisymmetric and transitive it is therefore an order relation.
A set R is said to be a proper subset of S if RcS but not R=S. We write this relation RcS. It follows that this relation is irreflexive, asymmetric and transitive, and is therefore a strict order relation.
Operations on sets
The set of all elements x such that xeA and xeB, denoted AnB, is called the intersection of A and B. Intersection is an idempotent, commutative and associative operation. For all A and B we have (AnB)cA. The relation AcB means the same as AnB = A. The largest set contained in A and B is their intersection AnB.
The set of all elements x such that xeA or xeB (or both) denoted AuB is called the union of A and B. Union is an idempotent, commutative and associative operation. For all A and B we have Ac(AuB). The relation AcB means the same as AuB = B. The smallest set containing A and B is their union AuB.
The set of all members x such that xeA but not xeB is a set, denoted A–B, called the difference of A and B. We can also form the difference B–A. The union of these two differences of A and B is called the symmetric difference of A and B, denoted A÷B = (A–B)u(B–A).
Full and Null Sets
The context within which a discussion takes place can be regarded as a set called the full set I. Within a given context there is also a uniquely defined null set O, that has no concepts from that context as members.
For any set A, within a given context, we have OcAcI. The complement of a set A is the difference I–A, also denoted –A. We have –I = O, –O = I, –(–A) = A, Au(–A) = I, An(–A) = O. If AcB then (–B)c(–A): note the reversal of order. The operations of intersection and union are duals of each other with respect to the complement operation: (–A)u(–B) = –(AnB) and (–A)n(–B) = –(AuB). We can express the difference of sets in terms of complement and intersection: A–B = An(–B).