A and B are two finite sets. The difference of the number of...

$\mathrm{A}$ and $\mathrm{B}$ are two finite sets. The difference of the number of elements of the power sets is 96 . $($ Assume $\mathrm{n}(\mathrm{A})>\mathrm{n}(\mathrm{B})$
Find $\mathrm{n}(\mathrm{A})+\mathrm{n}(\mathrm{B})$.
(1) 11
(2) 12
(3) 13
(4) 14

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Key Concepts

Diophantine Analysis

Diophantine analysis involves solving equations that require integer solutions. In the context of problems with finite sets and power sets, it is used to determine the specific integer values that satisfy equations derived from the exponential relationships.

Finite Sets

Finite sets are collections of distinct elements that can be completely counted. They form the basis of combinatorial techniques and are crucial in understanding how many items are in a group, which is often used in problems involving counting and subsets.

Power Set

A power set is the set of all subsets of a given set, including both the empty set and the set itself. For any finite set with n elements, the power set contains 2^n subsets since each element can either be included or excluded.

Exponential Equations

Exponential equations involve expressions where the variable appears as an exponent. When applied to power sets, these equations relate the number of elements in a set to 2 raised to that number, allowing us to compare the sizes of different power sets.

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