Sets and Functions Help (page 2)

Introduction to Sets and Functions

A set is a collection of objects. We denote a set with a capital roman letter, such as S or T or U. If S is a set and s is an object in that set then we write s S and we say that s is an element of S. If S and T are sets then the collection of elements common to the two sets is called the intersection of S and T and is written S ∩ T. The set of elements that are in S or in T or in both is called the union of S and T and is written S ∪ T.

A function from a set S to a set T is a rule that assigns to each element of S a unique element of T. We write f : S → T.

Examples

You Try It: Let f be the rule that assigns to each real number its cube root. Is this a function?

In calculus, the set S (called the domain of the function) and the set T (called the range of the function) will usually be sets of numbers; in fact they will often consist of one or more intervals in . The rule f will usually be given by one or several formulas. Many times the domain and range will not be given explicitly. These ideas will be illustrated in the examples below.

You Try It: Consider the rule that assigns to each real number its absolute value. Is this a function? Why or why not? If it is a function, then what are its domain and range?

Examples of Functions of a Real Variable

Example 1

Math Note: Notice that, in the definition of function, there is some imprecision in the definition of T . For instance, in Example 1.24, we could have let T = [0, ∞) or T = (−6, ∞) with no significant change in the function. In the example of the “name” function (Example 1.23), we could have let T be all strings of letters not exceeding 5000 characters in length. Or we could have made it all strings without regard to length. Likewise, in any of the examples we could make the set S smaller and the function would still make sense.

It is frequently convenient not to describe S and T explicitly.