The Joy of Sets—Anon.:

Where Bugs Go

An understanding of set theory can be very helpful in dealing with databases; indeed, as mentioned in Chapter 7, the relational model is directly based on certain aspects of that theory. In this appendix, therefore, I offer a brief tutorial on set theory basics.

I’ll begin with a definition:

**Definition:**A**set***S*is a collection of distinct**elements**, or**members**, such that, given an arbitrary object*x*, it can be determined whether or not*x*is contained in*S*(i.e., is an element of*S*).*Note the terminology:*A set is said to**contain**its members.Here’s an example:

{ 2 , 3 , 5 , 7 }

As this example suggests, the standard “on paper” representation of a set is as a commalist, enclosed in braces, of symbols denoting the elements of the set in question. Points arising:

Sets never contain duplicate elements. Thus, if duplicate symbols appear in the “on paper” representation (which by convention they usually don’t), they can simply be ignored. For example, the following—

{ 2 , 2 , 3 , 5 , 5 , 5 , 7 , 7 }

—is another possible, albeit unlikely, “on paper” representation of the set {2,3,5,7}.

Sets have no ordering to their elements. Thus, for example, the following—

{ 7 , 2 , 5 , 3 }

—is another possible “on paper” representation of that same set.

A set can contain no elements at all, in which case it’s the (unique)

*empty set*, written { }, or sometimes Ø.There’s another unique set called the

*universal set*, which is the set that contains ...

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