Let $ S $ be a set consisting of 10 elements. The number of tuples of the form $ (A, B) $ such that $ A $ and $ B $ are subsets of $ S $, and $ A \subseteq B $, is $\underline{\hspace{2cm}}$.

Question

cdquestions Admin · Accepted Answer

Let $ S $ have 10 elements. For each element of $ S $, there are three possible choices: - The element is not in $ B $ - The element is in $ B $ but not in $ A $ - The element is in both $ A $ and $ B $ Since $ A \subseteq B $, an element cannot be in $ A $ without being in $ B $. Thus, each element has exactly 3 independent choices. Therefore, the total number of such ordered pairs $ (A, B) $ is: $$ 3^{10} = 59049 $$ Final Answer: $$ \boxed{59049} $$

Let \( S \) be a set consisting of 10 elements.
The number of tuples of the form \( (A, B) \) such that \( A \) and \( B \) are subsets of \( S \), and \( A \subseteq B \), is \(\underline{\hspace{2cm}}\).

Show Hint

Correct Answer: 59049

Solution and Explanation

Top Questions on Sets

Questions Asked in GATE CS exam

Let \( S \) be a set consisting of 10 elements. The number of tuples of the form \( (A, B) \) such that \( A \) and \( B \) are subsets of \( S \), and \( A \subseteq B \), is \(\underline{\hspace{2cm}}\).