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 is a subset of T?

The correct answer is 729.

Step-by-Step Explanation:

We are looking for the number of ordered pairs of sets (S, T) such that S and T are subsets of the set {1, 2, 3, 4, 5, 6} and S ⊆ T (S is a subset of T).

Method 1: Analysis by Element Choices
Let the universal set be A = {1, 2, 3, 4, 5, 6}. For S to be a subset of T (S ⊆ T), every element in A must fall into one of three possible, mutually exclusive regions in a Venn diagram representation:
1. The element belongs to both S and T (since S ⊆ T, any element in S must also be in T).
2. The element belongs to T but does not belong to S.
3. The element belongs to neither S nor T.

Note that it is impossible for an element to belong to S but not to T, because that would violate the subset condition S ⊆ T.

Since there are 6 elements in the set {1, 2, 3, 4, 5, 6}, and we make a choice for each element independently:

$$\text{Total number of pairs (S, T)}=3\times 3\times 3\times 3\times 3\times 3=3^6$$

$$3^6=729$$

Method 2: Analysis from the Image (Inductive Pattern)
As shown in the attached image, we can analyze the problem by starting with smaller sets and looking for a pattern:

• For a 1-element set {1}:
If T = ∅, then S must be ∅ (since S ⊆ T). This gives 1 pair: (∅, ∅).
If T = {1}, then S can be ∅ or {1}. This gives 2 pairs: (∅, {1}) and ({1}, {1}).
Thus, for 1 element, the total number of pairs is 1 + 2 = 3 pairs, which is:

$$3^1=3$$

• For a 2-element set:
As described in the image, the pattern continues and the total number of pairs is:

$$3^2=9$$

• For an n-element set:
Following the pattern for 3, 4, and 5 elements as listed in the image, the total number of pairs for a set with n elements is:

$$3^n$$

Since our given set {1, 2, 3, 4, 5, 6} contains exactly n = 6 elements, the total number of satisfying pairs (S, T) is:

$$3^6=729$$