Final answer:
Considering the definitions of set complement, difference, union, and intersection, only option (c) A = {a, b, f} and B = {a, g, h} correctly satisfies the given set relationships, as it accounts for all provided set operations.
Step-by-step explanation:
The question involves the concepts of set theory specifically dealing with operations like complement, difference, union, and intersection of sets. Given the universal set U and subsets A and B, we have information about A' intersect B (A' n B), the difference of A and B (A - B), and the complement of the union of A and B ((A U B)'). To determine which option correctly represents sets A and B, we need to apply the definitions of these operations. Here's how we analyze the given information:
- A' n B = {a, g, h} suggests that these elements are not in A but are in B.
- A - B = {b, f} indicates these elements are in A but not in B.
- (A U B)' = {c, d} means these elements are in neither A nor B.
Considering the options provided:
- Option (a) is incorrect because if A = {a, g, h}, A' would not contain a, g, and h, thus A' n B cannot be {a,g,h}.
- Option (b) is incorrect because A = {a, b, f} would mean that A' does not contain a, b, or f, which contradicts A' n B = {a,g,h}.
- Option (c) is correct because A = {a, b, f} does not include g or h (present in A' n B), and B does not contain b or f (present in A - B). There is also no conflict with (A U B)' = {c,d}
- Option (d) is incorrect because A cannot be {c, d} as (A U B)' indicates c and d are in neither A nor B.
Therefore, the correct answer is option (c), where A = {a, b, f} and B = {a, g, h}.