In: Advanced Math

Which of the following sets are not well defined? Explain.

**a. **The set of wealthy school teachers

**b. **The set of great books

**c. **The set of natural numbers greater than 100

**d. **The set of subsets of \(\{1,2,3,4,5,6\}\)

**e. **The set \(\{x \mid x \neq x\) and \(x \in N\}\)

**An object is well-defined if there is no ambiguity about whether it belongs to a set**, i.e., the definition of a set allows us to always tell what is a member of the set and what is not.

**a **is not a well-defined set, because we don't know how to define wealthy.

**b **is also not a well-defined set, because we don't know how to classify a book as great or not great.

**c **is a well-defined set because if you are given a number, you just need to check if it is a natural number and if it is greater than 100.

**d **is well-defined because if given a set, we only need to check if it is a subset of { 1, 2, 3, 4, 5, 6 }.

**e **is well-defined because it is an empty set. An element is given and we can easily determine that it is not in the set.

As a result, the definitions of sets **a** and **b** are not well defined.

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