Question

d. which of the following describes the set of subsets of {a,b,c,d,e,f}? a. it is not well defined. each subset of {a,b,c,d,e,f} is itself a set, and it does not make sense to talk about a set of sets. b. it is not well defined. although there is a clear meaning of \subsets,\ it is not possible to tell whether a set is a subset of {a,b,c,d,e,f}. c. it is well defined. it is possible to list all of the subsets of {a,b,c,d,e,f}. d. it is well defined. the set {a,b,c,d,e,f} is a subset of itself. e. it is well defined. there is a clear meaning of \subsets.\ f. it is not well defined. there is no clear meaning of \subsets.\

Explanation:

Step1: Recall set definition

A set is well - defined if we can clearly determine whether an object belongs to it. For the set of subsets of \(\{a,b,c,d,e,f\}\), the concept of a subset is well - defined (a subset \(S\) of a set \(A\) means every element of \(S\) is an element of \(A\)). Also, even though the number of subsets of a set with \(n\) elements is \(2^n\) (here \(n = 6\), so \(2^6=64\) subsets), we can still determine what a subset is. The set \(\{a,b,c,d,e,f\}\) is a subset of itself (since every element of \(\{a,b,c,d,e,f\}\) is in \(\{a,b,c,d,e,f\}\)).

Step2: Analyze each option

• Option A: It does make sense to talk about the set of subsets of a set. A set can be an element of another set (in the case of a power set, the set of all subsets of a set \(A\) is called the power set of \(A\), and each subset of \(A\) is an element of the power set). So A is wrong.
• Option B: The concept of a subset is well - defined. We can clearly tell whether a set is a subset of \(\{a,b,c,d,e,f\}\) (by checking if all its elements are in \(\{a,b,c,d,e,f\}\)). So B is wrong.
• Option C: The set of subsets of \(\{a,b,c,d,e,f\}\) is well - defined, and while we can list all subsets (for a small set like this, we can use methods like listing all single - element subsets, two - element subsets, etc.), the key point is that the set is well - defined. But let's check other options.
• Option D: The set of subsets of \(\{a,b,c,d,e,f\}\) is well - defined (because the condition for being a subset is clear), and \(\{a,b,c,d,e,f\}\) is a subset of itself (since every element of \(\{a,b,c,d,e,f\}\) is in \(\{a,b,c,d,e,f\}\)). This option correctly states that the set is well - defined and that \(\{a,b,c,d,e,f\}\) is a subset of itself.
• Option E: The meaning of "subsets" is clear (a subset of a set \(A\) is a set whose elements are all in \(A\)), so E is wrong.
• Option F: The set of subsets of \(\{a,b,c,d,e,f\}\) is well - defined, so F is wrong.

Answer:

D. It is well defined. The set \(\{a,b,c,d,e,f\}\) is a subset of itself.