Write true (T) or false (F):

(i) The number of proper subsets of a set containing n elements is 2ⁿ.

(ii) Any set A and its complement are equivalent sets.

(iii) The complement of a set is a subset of U.

(iv) If n(A ∩ B) = Φ, then n(B - A) = n(B)

(v) If two sets A and B are disjoint, then n(A ∪ B) = n(A) + n(B)

(i) False
Reason — The total number of subsets of a set containing n elements is 2ⁿ. However, a proper subset must be smaller than the set itself (it cannot be the set itself). Therefore, the number of proper subsets is 2ⁿ-1.

(ii) False
Reason — For two sets to be equivalent, they must have the same number of elements (n(A) = n(A')). This is only true if the set A contains exactly half the elements of the Universal set U. In most cases, the number of elements in a set and its complement are different.

(iii) True
Reason — By definition, the complement of a set A (A') consists of all elements that are in the Universal set (U) but not in A. Since every element of A' is an element of U, A' is a subset of U (A' ⊆ U).

(iv) True
Reason — The term n(B - A) represents the elements in B that are not in A. The formula is n(B) - n(A ∩ B). If n(A ∩ B) = 0 (meaning the sets are disjoint), then n(B) - 0 = n(B).

(v) True
Reason — For any two sets, n(A ∪ B) = n(A) + n(B) - n(A ∩ B). If sets A and B are disjoint, their intersection is empty (n(A ∩ B) = 0), which simplifies the formula to n(A ∪ B) = n(A) + n(B).