Question

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

Explanation:

To determine the set of subsets of \(\{a, b, c, d\}\), we recall the definition of a subset: a set \(S\) is a subset of \(T\) if every element of \(S\) is in \(T\). The set of subsets (power set) of a finite set with \(n\) elements has \(2^n\) elements, and we can list them (e.g., \(\emptyset\), \(\{a\}\), \(\{b\}\), \(\{c\}\), \(\{d\}\), \(\{a,b\}\), etc.). The meaning of "subset" is clear, and we can determine membership. Option C states it's well - defined (clear meaning of "subsets") and we can list all subsets (since the set is finite, \(n = 4\), so \(2^4=16\) subsets, which is manageable to list). Option A is wrong as we can determine subset membership. Option B is wrong as "subset" is well - defined. Option D: while we can list all subsets (for a 4 - element set), the key is the set of subsets is well - defined. Option E is wrong as subsets of a set are well - defined (each subset is a set where elements are from the original set). Option F is about the set being a subset of itself, which is true, but the question is about the set of subsets being well - defined. So the correct description is that the set of subsets of \(\{a,b,c,d\}\) is well - defined (clear meaning of "subsets").

Answer:

C. It is well defined. There is a clear meaning of "subsets."