Lesson 5 in Perceptive Mathematics: union of sets

If you love collecting shells from a beach, you surely have at your home shells collected from several different beaches. 

Let's say you have shells from three different beaches. Let B1 be the set of shells collected from the first beach, B2 the set of shells collected from the second beach, and B3 the set of shells collected from the third beach. Altogether, your collecting activity can be recorded by the set A={B1, B2, B3}. However, when you bring those shells at home, you will most likely mix them all up in one heap of shells. So if for instance, B1={S1, S2, S3}, B2={S4, S5} and B3={S6, S7, S8, S9}, where the symbols S1, ...., S9 represent the collected shells, then your collection at home, after the shells have been placed all together, is given by the set C={S1, S2, ..., S9}. Now, more generally, the union of a set A is the set C of all elements of individual elements of A. We write the union of A as UA, thus C=UA.

Exploration 5.1. Compute the following unions (i.e., write down what set they equal to by explicitly listing the elements enclosed between braces).  

• U{}=?
• U{{}}=?
• U{{{}}}=?
• U{{},{A}}=?
• U{{A, B}, {A, {}}, {C}}=?
• U{{A, B}, {A, C}, {{A, C, {}}}}=?

The union of a set can be seen as a reverse construction both to the singleton construction as well as the quotient construction. If we begin with a set A, then take its singleton, {A}, and then take the union, U{A}, we get back the same set A. Indeed, the elements of U{A} must be elements of elements of {A}. But {A} has only one element, it is A. So elements of U{A} are the same as elements of A, which means that the two sets are the same set. Similarly, if Q is a quotient of the set A, then elements of UQ are all elements of classes formed during partitioning. We know by the first axiom of partitioning that these elements must be elements of A. Moreover, by the second axiom of partitioning, no element of A will be missed (every element of A is an element of some class, i.e., of some element of Q). So the sets UQ and A are once again the same set. Intuitively, if partitioning separates a set into pieces, the union puts these pieces back together.

Exploration 5.2. 

• Draw several different tree diagrams for sets, ensuring in each case that every element of the set being considered is itself a set. Then, for each of these drawings, draw the tree diagram for the union of the corresponding set. 
• While busy with the task above, did you notice that the union construction shrinks the height of the tree? In contrast, the singleton and the quotient constructions grow the trees (make a few sketches to get convinced of this).