What is a Well-Defined Set?

A well-defined set is a set for which it is clear and unambiguous whether any given object belongs to the set or not. In other words, the criteria or rule that determines membership in the set must be precise and leave no room for doubt.

Key Points:

• Clear membership criteria: For every element, you can definitively say "yes" or "no" to whether it is in the set.
• No ambiguity: The description or rule that defines the set must be specific enough to avoid confusion.
• Examples of well-defined sets:
  • The set of all even numbers.
  • The set of all countries in Europe.
  • The set of all prime numbers less than 50.

Contrast with Not Well-Defined Sets:

A set that is not well-defined might have vague or subjective criteria, making it unclear whether some elements belong to it. For example:

• The set of "tall people" — because "tall" is subjective and can vary from person to person.
• The set of "beautiful paintings" — because beauty is subjective.

Summary:

A well-defined set has a clear rule or property that allows anyone to determine membership without ambiguity, ensuring the set is logically consistent and meaningful in mathematics or any formal context.