What Is a Well-Defined Set in Mathematics?

In mathematics, a well-defined set clearly indicates what is a member of the set and what is not. For example, a set that is identified as “the set of even whole numbers between 1 and 11” is a well-defined set because it is possible to identify the exact members of the set: 2, 4, 6, 8 and 10. There is only one possible solution set that fits this description.

Another example of a well-defined set is “the set of integers from -3 to 3, inclusive.” This set clearly contains -3, -2, -1, 0, 1, 2 and 3 and only those integers.

On the other hand, “the set of lucky numbers” is not well-defined because it is open to interpretation. It is not clear from the description what “lucky” means, whose lucky numbers will be considered and what has to happen for a number to be considered “lucky.” There are any number of possible solution sets.

In set theory, it does not matter how the members of the set are arranged. Therefore, {2, 3, 4, 6, 8, 10} is identical to {10, 2, 4, 8, 6, 3}. It does not affect the well-defined nature of a set.

When discussing sets, the “order” of the sets does not refer to their arrangement but to their sizes. “The set of even whole numbers between 1 and 11” has an order of 6.