Webwhompers

Explicit set notation is convenient for small sets but impractical for large sets, such as the set of all Web pages. Implicit set notation (also called set builder notation) is a more sophisticated way to describe sets with mathematical rigor.

There are two parts to any implicit set expression:

1. A dummy variable. Very often we will use x as our dummy variable.
2. A logical true/false statement. Usually, this statement is an expression that uses the dummy variable.

These two parts are enclosed within curly braces and separated by a colon (":"). For example:

• A = {x : x is a single-digit positive integer}
• B = {x : x is an even element of set A}

The above two sets can also be written explicitly:

• A = {1,2,3,4,5,6,7,8,9}
• B = {2,4,6,8}

How to interpret implicit set notation

Suppose P(x) is any logical true/false statement that uses x in some way. (For example, P(x) = "x is a single-digit positive integer"; or P(x) = "x is a Web page".) With that, we can write implicit set notation that looks like this: {x : P(x)}. The set defined by {x : P(x)} is the set of all objects for which logical statement P is true. For example:

1. Suppose P(x) = "x is a Web page"; then
2. {x: P(x)} = {x: x is a Web page}
3. http://brucehoppe.com/img/index.html ∈ {x: x is a Web page}
4. Bruce Hoppe ∉ {x: x is a Web page}
5. |{x: x is a Web page}| is approximately 50 billion

Statement #3 above is true because "http://brucehoppe.com/img/index.html is a Web page" is true. Similarly, statement #4 above is true because "Bruce Hoppe is a Web page" is false.