To say that sets A and B are equal means that they have the same elements, i.e. A = B means ‘m ∈ A implies and is implied by m ∈ B’.
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 R ⊆ S. From the definition it follows that any set is a subset of itself, that is S ⊆ S, that is the relation ⊆ is reflexive. If R ⊆ S and S ⊆ R then it follows that R = S, that is the relation ⊆ is antisymmetric. If R ⊆ S and S ⊆ T then it follows that R ⊆ T, that is the relation ⊆ 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 R ⊆ S but R ≠ S. We write this relation R ⊂ S. It follows that this relation is irreflexive, asymmetric and transitive, and is therefore a strict order relation.
The set of all elements x such that x ∈ A and x ∈ B, denoted A ∩ B, is called the intersection of A and B. Intersection is an idempotent, commutative and associative operation. For all A and B we have (A ∩ B) ⊆ A. The relation A ⊆ B means the same as A ∩ B = A. The largest set contained in A and B is their intersection.
The set of all elements x such that x ∈ A or x ∈ B (or both) denoted A ∪ B is called the union of A and B. Union is an idempotent, commutative and associative operation. For all A and B we have A ⊆ (A ∪ B). The relation A ⊆ B means the same as A ∪ B = B. The smallest set containing A and B is their union.
The set of all members x such that x ∈ A but not x ∈ B 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) ∪ (B − A).
Within a given context we can define a full set I and a null set O. Then for any set A in that context we have O ⊆ A ⊆ I. The complement of a set A is the difference I − A, also denoted in the operator form −A. We then have −I = O, −O = I, −(−A) = A, A ∪ −A = I, A ∩ −A = O.
If A ⊆ B then −B ⊆ −A, note the reversal of order.
The operations of intersection and union are duals of each other with respect to the complement operation: −A ∪ −B = −(A ∩ B) and −A ∩ −B = −(A ∪ B).
We can express the difference of sets in terms of complement and intersection: A − B = A ∩ −B.
The algebra of all the subsets of a given set provide an important example of a Boolean algebra. The order relation is that of one set being a subset of another, A ⊆ B. The full set I and the null set O are the first and last elements. Intersection and Union are the dual operations.