Difficulty: Hard

Average Score: 37%

Let A and B be disjoint sets, and let C be such that C∩ A and C ∩ B are both nonempty. Which of the following statements is true?

Rationale

A and B ∩ C are disjoint.

Since sets A and B are disjoint, they do not share any elements. When we take the intersection of B with C, which is nonempty, the intersection will not introduce any elements from A, ensuring that A remains disjoint from the intersection of B and C.

A (A ∩ C and B ∩ C are disjoint) YOUR ANSWER

This statement is incorrect because A ∩ C is the intersection of set A with set C, while B ∩ C is the intersection of set B with set C. Since C is a nonempty subset of both A and B, there can be elements in C that are common to both intersections, thus they are not necessarily disjoint.

B (A∩ C and B ∩ C are disjoint) YOUR ANSWER

This choice is also incorrect for the same reason as A). The nonempty nature of C implies that it can contain elements from both A and B. Therefore, A ∩ C and B ∩ C can both contain elements from C, making them potentially overlapping, contradicting the statement that they are disjoint.

C (A and B ∩ C are disjoint) CORRECT

This statement is true because A and B are disjoint sets by definition. Since B ∩ C is merely a subset of B, it cannot contain any elements from A, maintaining the disjoint property between A and B ∩ C.

D (None of the above) YOUR ANSWER

This choice is incorrect as option C is indeed a true statement. Therefore, the assertion that none of the options are true is false.

Conclusion

In summary, while sets A and B are disjoint, the intersection of B with the nonempty set C does not affect their disjoint status. Therefore, A remains disjoint from B ∩ C. The other options incorrectly imply potential overlaps between the intersections and sets, which is not consistent with the properties of disjoint sets.

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