11. If [tex]A={2,3,4}, B={3,5,6}[/tex], and [tex]c={5,7,9}[/tex], prove the associative property of the union of set.
12. If [tex]A={m, n, o}, B={p, q, r, s}[/tex], and [tex]C={n, o, p, q}[/tex], prove the associative property of intersection of sets.

For the union of sets A = { 2 , 3 , 4 } , B = { 3 , 5 , 6 } , and C = { 5 , 7 , 9 } , we have ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) = { 2 , 3 , 4 , 5 , 6 , 7 , 9 } .
For the intersection of sets A = { m , n , o } , B = { p , q , r , s } , and C = { n , o , p , q } , we have ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) = { } .
Thus, the associative property holds for both union and intersection for the given sets.
The final answer is that the associative property is proven for both the union and intersection of the given sets: Associative property holds ​ .

Explanation

Understanding the Problem We are given three sets: A = { 2 , 3 , 4 } , B = { 3 , 5 , 6 } , and C = { 5 , 7 , 9 } . We need to prove the associative property of the union of these sets, which states that ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) .

Finding A union B First, let's find A ∪ B . This is the set of all elements that are in A or B or both. So, A ∪ B = { 2 , 3 , 4 , 5 , 6 } .

Finding (A union B) union C Now, let's find ( A ∪ B ) ∪ C . This is the set of all elements that are in A ∪ B or C or both. So, ( A ∪ B ) ∪ C = { 2 , 3 , 4 , 5 , 6 } ∪ { 5 , 7 , 9 } = { 2 , 3 , 4 , 5 , 6 , 7 , 9 } .

Finding B union C Next, let's find B ∪ C . This is the set of all elements that are in B or C or both. So, B ∪ C = { 3 , 5 , 6 } ∪ { 5 , 7 , 9 } = { 3 , 5 , 6 , 7 , 9 } .

Finding A union (B union C) Now, let's find A ∪ ( B ∪ C ) . This is the set of all elements that are in A or B ∪ C or both. So, A ∪ ( B ∪ C ) = { 2 , 3 , 4 } ∪ { 3 , 5 , 6 , 7 , 9 } = { 2 , 3 , 4 , 5 , 6 , 7 , 9 } .

Conclusion for Union We have found that ( A ∪ B ) ∪ C = { 2 , 3 , 4 , 5 , 6 , 7 , 9 } and A ∪ ( B ∪ C ) = { 2 , 3 , 4 , 5 , 6 , 7 , 9 } . Since these two sets are equal, we have proven the associative property of the union of sets for the given sets A , B , and C .

Understanding the Problem for Intersection Now, let's move on to the second part of the problem. We are given three sets: A = { m , n , o } , B = { p , q , r , s } , and C = { n , o , p , q } . We need to prove the associative property of the intersection of these sets, which states that ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) .

Finding A intersection B First, let's find A ∩ B . This is the set of all elements that are in both A and B . So, A ∩ B = { m , n , o } ∩ { p , q , r , s } = { } (empty set) .

Finding (A intersection B) intersection C Now, let's find ( A ∩ B ) ∩ C . This is the set of all elements that are in both A ∩ B and C . Since A ∩ B is an empty set, ( A ∩ B ) ∩ C = { } ∩ { n , o , p , q } = { } (empty set) .

Finding B intersection C Next, let's find B ∩ C . This is the set of all elements that are in both B and C . So, B ∩ C = { p , q , r , s } ∩ { n , o , p , q } = { p , q } .

Finding A intersection (B intersection C) Now, let's find A ∩ ( B ∩ C ) . This is the set of all elements that are in both A and B ∩ C . So, A ∩ ( B ∩ C ) = { m , n , o } ∩ { p , q } = { } (empty set) .

Conclusion for Intersection We have found that ( A ∩ B ) ∩ C = { } (empty set) and A ∩ ( B ∩ C ) = { } (empty set) . Since these two sets are equal, we have proven the associative property of the intersection of sets for the given sets A , B , and C .


Examples
Understanding set operations like union and intersection is crucial in computer science, especially in database management and data analysis. For example, if you have two lists of customers, one who bought product A and another who bought product B, the union of these sets gives you all customers who bought either product A or B, while the intersection gives you customers who bought both. Associative properties ensure that combining multiple lists in any order yields the same result, which is vital for consistent data processing.

Answered by GinnyAnswer | 2025-07-08

We proved the associative property of union for sets A, B, and C by showing that ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) . Similarly, for the intersection of sets A, B, and C, we demonstrated that ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) . Both properties hold true for the provided sets.
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Answered by Anonymous | 2025-07-13