Solution

The correct answer is A - I, B - II, C - IV, D - III

EXPLANATION:

• In the context of Set Theory:
  • ϕ denotes the empty set, also written as {}.
  • ∩ denotes intersection, which means it gives the common elements in both sets.
  • ∪ denotes union, which means it gives all unique elements from both sets.
  • . denotes set difference, which means it gives elements of first set that are not in the second set.

Now, let's evaluate the set operations:

• A. ϕ ∩ {ϕ} = ϕ : Here the intersection of the empty set and a set consisting of the empty set is the empty set itself as there are no common elements. That means A should match with I.
• B. {ϕ} ∩ {ϕ} = {ϕ} : The intersection of a set consisting of the empty set and itself is the set with the empty set. That means B should match with II.
• C. {ϕ, {ϕ}} - ϕ = {ϕ, {ϕ}} : Here, the set difference give same set. So, C should match with IV.
• D. ϕ ∪ {{ϕ}} = {{ϕ}} : The union of the empty set and the set of the empty set is just the set containing the empty set. So, D should match with III.

Given these matching pairs, the correct answer from the given choices would be: A - I, B - II, C - IV, D - III