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Chapter 4 of "Naive Set Theory"

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Suppose we want to unite the elements of two sets into one comprehensive set. We require a specific notation in order to combine sets, as well as specific notation to specify which sets are being combined. The combination of the elements of multiple sets into one comprehensive set is known as a "union." Unions are made possible in set theory thanks to the "Axiom of Unions." Halmos describes the Axiom of Unions thus: "For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection." Thus, when we speak of a union, we are speaking of a union of a "collection" of sets, represented, for our purposes, with a capital C. Union is represented with ∪. Thus, we speak of the union of A and B as A ∪ B.

We articulate the Axiom of Unions with the help of first-order logic thus:

Axiom of Unions: ∀x∃y∀z[z∈y ≡ ∃w(w∈x & z∈w)]

We might paraphrase this formal articulation like this: "Every z is a member of at least one y, if and only if there is at least one w that is a member of every x and every z is a member of at least one w." Thus, we have the logical foundation for our axiom of unions.

In set-builder notation, Halmos recommends the following:

U = {x: x ∈ X for some X in C}

The union = x is such that x is a member of X for some X in C. Provided x is an element of at least one set in the collection of sets C, we're good to go.

We also have intersection, represented by ∩. An intersection refers to the occurrence of an element in both sets, rather than at least one, as we saw was the case with unions. We say that the intersection of two sets is disjoint when they have nothing in common. A more technical way of saying this is to say that the intersecftion of any two such sets is an empty set. We would represent this as follows:

A ∩ B = Ø

The following properties inhere in the operations of union and intersection:

Distributivity:

A U (B ∩ C) = (A U B) ∩ (A U C)

A ∩ (B U C) = (A ∩ B) U (A ∩ C)

Commutativity:

A U B = B U A

Associativity:

A U (B U C) = (A U B) U C

Halmos, Paul. "Naive Set Theory." Litton Educational Publishing, 1960.

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