Determining the Number of Possible Sets A Given Intersection and Union

Given the sets A and B with the conditions A ∩ B {1, 2, 3, 4} and A ∪ B {1, 2, 3, 4, 5, 6, 7}, we need to determine how many different sets are possible for A. This problem involves understanding the relationship between the intersection and union of sets. We will break down the solution into multiple steps to ensure clarity and accuracy.

Identifying the Elements in A and B

The intersection of A and B, denoted by A ∩ B {1, 2, 3, 4}, indicates that the elements 1, 2, 3, and 4 are common to both sets A and B. The union of A and B, denoted by A ∪ B {1, 2, 3, 4, 5, 6, 7}, includes all elements present in either A or B or both. This gives us the additional elements 5, 6, and 7, which must be either in A or B or both, but not necessarily in both.

Options for the Elements 5, 6, and 7

For each of the elements 5, 6, and 7, there are two possible scenarios:

The element can be in set A. The element can be in set B.

Since the elements 5, 6, and 7 cannot be in both sets A and B due to the given conditions, each element has exactly 2 choices.

Calculating the Number of Combinations

For three elements (5, 6, and 7), the total number of different combinations is:

[text{Total combinations} 2^3 8]

This means there are 8 different ways to assign the elements 5, 6, and 7 to sets A and B.

Conclusion

Each unique combination of the elements 5, 6, and 7 results in a distinct set A. Therefore, the total number of different possible sets A is 8.

Simplified Explanation of Possible Sets A

Since the intersection of A and B includes the elements 1, 2, 3, and 4, the set A must contain these elements. Additionally, the elements 5, 6, and 7 can independently be part of A or B. The following combinations represent the different possible sets A:

A {1, 2, 3, 4} A {1, 2, 3, 4, 5} A {1, 2, 3, 4, 5, 6} A {1, 2, 3, 4, 5, 6, 7}

These combinations cover all possible sets A, given the constraints of the problem.

Combinatorial Analysis

The combinatorial analysis can also be performed by considering the different ways the elements 5, 6, and 7 can be distributed:

No elements from {5, 6, 7} in A: 1 way. Exactly one element from {5, 6, 7} in A: 3 ways. Exactly two elements from {5, 6, 7} in A: 3 ways. All three elements from {5, 6, 7} in A: 1 way.

Add these up to get: 1 3 3 1 8 different sets A.

Conclusion

The total number of different sets A is boxed{8}, as each combination of the elements 5, 6, and 7 results in a distinct set A.