Explanation
Step 1: Analyze the given conditions
We are given:
A∩C=ϕ
B∩C=ϕ
A∪C=B∪C
Step 2: Evaluate the conditions
To check if A=B, we can use the distributive property of sets on the third condition:
(A∪C)∩(A∪B)=(B∪C)∩(A∪B)
Alternatively, let's intersect both sides of A∪C=B∪C with Cc (assuming a universal set U where Cc=U−C):
(A∪C)∩Cc=(B∪C)∩Cc
(A∩Cc)∪(C∩Cc)=(B∩Cc)∪(C∩Cc)
(A−C)∪ϕ=(B−C)∪ϕ
A−C=B−C
Since we are given A∩C=ϕ and B∩C=ϕ, it implies A⊆Cc and B⊆Cc. Therefore, A−C=A and B−C=B.
Thus, A=B. Statement II is true.
Step 3: Check Statements I and III
Regarding I: Does C=ϕ necessarily? No. C can be any set that is disjoint from A and B. For example, if A={1},B={1}, and C={2}, the conditions A∩C=ϕ and B∩C=ϕ are satisfied, and A∪C={1,2}=B∪C. Here C=ϕ. Statement I is false.
Regarding III: A∪B=C? In our example above, A∪B={1} and C={2}. Clearly, A∪B=C. Statement III is false.
Conclusion:
Only Statement II is necessarily true.
Final Answer:
The correct option is (b) II only.
