1. **State the problem:** We are given the sizes of sets and their union and need to find the size of the intersection of sets A and B. 2. **Given:** - $n(U) = 143$ (size of universal set) - $n(A) = 43$ - $n(B) = 86$ - $n(A \cup B) = 129$ 3. **Formula used:** The formula relating union and intersection of two sets is: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ 4. **Rearrange to find $n(A \cap B)$:** $$n(A \cap B) = n(A) + n(B) - n(A \cup B)$$ 5. **Substitute the values:** $$n(A \cap B) = 43 + 86 - 129$$ 6. **Calculate:** $$n(A \cap B) = 129 - 129$$ $$n(A \cap B) = 0$$ 7. **Interpretation:** The intersection of sets A and B is empty, meaning they have no elements in common.