If A is any set, then

A. A∪A′=ϕ

B. A∪A′=U

C. A∩A′=U

D.None of these

Let's break down the given question and evaluate each option step by step.

We are given a set A and its complement A', where A' represents all the elements not in set A.

Option A: A ∪ A' = ϕ
The union of a set A and its complement A' represents all the elements that belong either to A or to A' (or to both). Since A' contains all elements not in A, the union of A and A' will cover all possible elements in the universal set U. Therefore, the union A ∪ A' is equal to U (the universal set), not the empty set ϕ. Thus, A ∪ A' = ϕ is false.

Option B: A ∪ A' = U
As mentioned above, the union of A and A' will cover every element in the universal set U because A' contains all elements not in A, and A contains all elements in A. Hence, A ∪ A' = U is true.

Option C: A ∩ A' = U
The intersection of A and its complement A' represents the elements that are in both A and A'. Since no element can belong to both A and A' (because A' contains all elements not in A), the intersection of A and A' is always the empty set ϕ, not the universal set U. Therefore, A ∩ A' = U is false.

Option D: None of these
This option is incorrect because we have already determined that option B is true.

Final Answer:
The correct answer is B. A ∪ A' = U.