Question

If $A_1, A_2, A_3, ... A_{100}$ are sets such that $n(A_i)=i+2; A_1 \subset A_2 \subset A_3 ... \subset A_{100}$ and $\bigcap_{i=3}^{100} A_i = A_n$ then n(A) is equal to

• A. 3
• B. 4
• C. 5
• D. 16

Answer 1

Answer: (C)

5

Answer 2

The problem states that $A_1 \subset A_2 \subset A_3 \subset ... \subset A_{100}$, which means that each set is a subset of the following set. This also implies that $A_3 \subset A_4 \subset ... \subset A_{100}$.

The intersection $\bigcap_{i=3}^{100} A_i$ includes all the sets from $A_3$ to $A_{100}$. Since they are nested subsets, the intersection is the smallest set, which is $A_3$. Therefore, $\bigcap_{i=3}^{100} A_i = A_3$.

We are given that $\bigcap_{i=3}^{100} A_i = A_n$. Thus, $A_n = A_3$, which implies $n = 3$.

We want to find $n(A)$, which is the cardinality of set A. Since $\bigcap_{i=3}^{100} A_i = A_n$, it is implied that $A = A_n$. Therefore, $A = A_3$.

The cardinality of $A_i$ is given by $n(A_i) = i + 2$. So, $n(A_3) = 3 + 2 = 5$.

Therefore, $n(A) = 5$.