A set is a collection of objects called elements of the set. Why not use the word ``collection'' and eliminate the word ``set'', thereby having fewer words to worry about? ``Collection'' is a common word whose generic meaning is understood by most people. The use of the word ``set'' means that there is also a method to determine whether or not a particular object belongs in the set. We then say that the set is well-defined. For example, it is easy to decide that the number 8 is not in the set consisting of the integers 1 through 5. After all, there are only five objects to consider and it is clear that 8 is not one of them by simply checking all five.
A basic problem here is now to indicate sets on paper and verbally. As seen above, a set could be described with a phrase such as ``the integers 1 through 5'' and the speaker hopes that it is understood. Symbollically, we use two common methods to write sets. The roster notation is a complete or implied listing of all the elements of the set. So and are examples of roster notation defining sets with 4 and 20 elements respectively. The ellipsis, `` '', is used to mean you fill in the missing elements in the obvious manner or pattern, as there are too many to actually list out on paper. The set-builder notation is used when the roster method is cumbersome or impossible. The set B above could be described by . The vertical bar, ``|'', is read as ``such that'' so this notation is read aloud as ``the set of x such that x is between 2 and 40 (inclusive) and x is even.'' (Sometimes a colon is used instead of |.) In set-builder notation, whatever comes after the bar describes the rule for determining whether or not an object is in the set. For the set the roster notation would be impossible since there are too many reals to actually list out, explicitly or implicitly.
To discuss and manipulate sets we need a short list of symbols commonly used in print. We start with five symbols summarized in the following table.
The first symbol, , indicates membership of an object in a particular set. The negation of this, or nonmembership is often indicated by `` '' (``x is not in A''). The subset relation, , states that every element of A is also an element of B. Logically, this would be: if then The union and intersection operators form new sets by the following rules. The set is defined to be while is defined to be . Finally, the complement of a set consists of those objects that are not in the given set. This presents a minor problem. If then clearly I am not in A so should I be considered an element of ? Normally not, I think. Underlying a discussion or argument involving sets is usually a large set called the universal set or universe of the discourse and is commonly denoted by U. This universe may be implied or stated explicitly. Operations involving union, intersection or complement are understood to be contained in this universe. For example, if we were discussing real numbers (so that our universe would be the set of reals) and mentioned the set A above with 3 elements, it is understood that consists of those real numbers not in A. This conveniently excludes me from the set .