A well-defined set is a collection of objects or elements for which it is always clear whether a given object belongs to the set or not, without any ambiguity. In other words, the criteria for membership in the set must be precise and unambiguous so that anyone can determine definitively if an object is in the set
Key points about well-defined sets:
- There is no confusion or uncertainty about the membership of any object.
- The defining property or rule for the set must be clear and specific.
- Examples of well-defined sets include:
- The set of primary colors: {red, blue, yellow}
- The set of even integers from 0 to 20
- The set of currently enrolled students at a university with a GPA above 3.0
- The set of residents of Minnesota aged between 18 and 25 years inclusive
Examples of sets that are not well-defined:
- The set of "good students" at a school (because "good" is subjective)
- The set of "young people" in a state (because "young" is ambiguous)
- The set of "best songs" or "favorite restaurants" (because these depend on personal opinion)
Thus, a well-defined set ensures clarity and consistency in membership, which is fundamental in set theory and mathematics.
