Question 9

A and B and C are disjoint

The statement A and B and C are disjoint is true because A and B are disjoint sets, meaning they do not share any elements, and the intersections C ∩ A and C ∩ B are nonempty, which implies that C intersects both sets without causing any overlap between A and B.

A) A ∩ C and B ∩ C are disjoint

This option is incorrect because A ∩ C and B ∩ C can have elements from set C that are distinct but do not overlap with each other. Since A and B are disjoint, it does not imply that their intersections with C must also be disjoint.

B) A∩ C and B ∩ C are disjoint

This option is similar to A and is incorrect for the same reason. The intersections of C with A and B can contain elements that are common to C but do not intersect with one another due to the disjoint nature of A and B.

C) A and B ∩ C are disjoint

This statement is correct because since A and B are disjoint sets, there are no common elements between A and B, and hence any intersection of A with elements of C does not overlap with any intersection of B with elements of C.

D) None of the above

This option is incorrect since statement C is valid. Thus, it is not true that none of the statements hold; in fact, one of them is indeed true.

Conclusion

The correct answer is C because it accurately describes the relationship between the sets A, B, and their intersections with C. Options A and B incorrectly suggest that the intersections with C are disjoint when they may not be, while D dismisses the validity of C, which is established as true. Therefore, C is the definitive choice based on the properties of disjoint sets.