Question

In: Advanced Math

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

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 \(x \in N\}\)

 

Solution

Step #1

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.