Operations On Sets

We are given the identity: $ A / B (A / B) / C = (A \setminus B) \setminus C $

Let be a set with elements.

We want to count the number of ways to choose two subsets and such that:

• The number of elements in is exactly one more than the number of elements in

Step 1: Fix the size of

Let the size of be , where can range from to . (We stop at because must have elements, and cannot have more than elements.)

Step 2: Choose

There are ways to choose a subset with elements from the elements of .

Step 3: Choose such that and

We must choose 1 element from the remaining elements (elements in but not in ) to add to to form .

This can be done in ways.

Total number of such pairs for fixed :

Step 4: Sum over all valid values of (from to ):

This sum simplifies to:

This is a known identity in combinatorics. Hence,

Final Answer: Option (A):