Question

How many pairs of sets \((S, T)\) are possible among the subsets of \(\{1,2,3,4,5,6\}\) that satisfy the condition that \(S \subseteq T\)?

• A. 729
• B. 728
• C. 665
• D. 664

Hint:

When counting subsets with \(S \subseteq T\), think element-wise. Each element has 3 options (in T only, in both S and T, or in neither).

Answer 1

The Correct Option is A

Step 1: Set Definition and Subset Count.
The given set comprises 6 elements: \(\{1,2,3,4,5,6\}\). The total number of possible subsets for this set is \(2^6 = 64\).

Step 2: Subset Condition \(S \subseteq T\).
Each element can be assigned to one of three categories relative to sets \(S\) and \(T\): 1. The element belongs to \(T\) exclusively.
2. The element is a member of both \(S\) and \(T\).
3. The element is absent from both \(S\) and \(T\). Consequently, each of the 6 elements has 3 distinct valid placements.

Step 3: Calculation of Total Pair Combinations.
\[\text{Total pairs} = 3^6 = 729\] \[\boxed{729}\]