How many combinations of non-null sets A, B, C are possible from the subsets of {2, 3, 5} satisfying the conditions: (i) A is a subset of B, and (ii) B is a subset of C?

Steps to Solve

1. Identify the set and its subsets The given set is $S = {2, 3, 5}$. The total number of subsets of a set with $n$ elements is $2^n$.

For set $S$, since there are 3 elements: $$ \text{Total subsets} = 2^3 = 8. $$

2. List the possible subsets The subsets of the set $S$ are:

• $\emptyset$ (the empty set)
• ${2}$
• ${3}$
• ${5}$
• ${2, 3}$
• ${2, 5}$
• ${3, 5}$
• ${2, 3, 5}$

Since we are looking for non-empty subsets, we will have 7 valid subsets.

3. Define the relationships between sets A, B, and C We need to satisfy:

• $A \subseteq B$
• $B \subseteq C$

4. Break down the selection process for each combination

• For each element in $S$, choose whether it belongs to $A$, $B$, or $C$:
  • If an element is in $A$, it automatically goes into $B$ and $C$.
  • If it is in $B$ but not in $A$, it goes into $C$.

5. Count the valid arrangements For each of the 3 elements in the set, there are 3 options:

• Not included
• Included in $C$ (and potentially $B$)
• Included in both $B$ and $A$

Thus, for $n=3$: $$ \text{Total combinations} = 3^3 = 27. $$

6. Exclude invalid combinations We must exclude the scenarios where at least one set, $A$, $B$, or $C$, is empty:

• Calculate when one is empty:
  • If $A$ is empty, we cannot choose non-empty sets $B$ and $C$.

No valid feasible combinations can arise from an empty $A$, so those configurations don't contribute.

The final tally confirms the combination possibilities excluding empty sets remain valid.