Naive Set Theory - Unions, Intersections, and Relative Complements

Unions, Intersections, and Relative Complements

Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). It is denoted by A ∪ B.

The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B.

Finally, the relative complement of B relative to A, also known as the set theoretic difference of A and B, is the set of all objects that belong to A but not to B. It is written as A \ B or A − B.

Symbolically, these are respectively

A ∪ B := {x : (x ∈ A) or (x ∈ B)};

A ∩ B := {x : (x ∈ A) and (x ∈ B)} = {x ∈ A : x ∈ B} = {x ∈ B : x ∈ A};

A \ B := {x : (x ∈ A) and not (x ∈ B) } = {x ∈ A : not (x ∈ B)}.

Notice that A doesn't have to be a subset of B for B \ A to make sense; this is the difference between the relative complement and the absolute complement from the previous section.

To illustrate these ideas, let A be the set of left-handed people, and let B be the set of people with blond hair. Then A ∩ B is the set of all left-handed blond-haired people, while A ∪ B is the set of all people who are left-handed or blond-haired or both. A \ B, on the other hand, is the set of all people that are left-handed but not blond-haired, while B \ A is the set of all people who have blond hair but aren't left-handed.

Now let E be the set of all human beings, and let F be the set of all living things over 1000 years old. What is E ∩ F in this case? No living human being is over 1000 years old, so E ∩ F must be the empty set {}.

For any set A, the power set is a Boolean algebra under the operations of union and intersection.