Let $D=\{16,19,21\}, E=\{16,18,19,20\}$ and $F=\{15,17,18,19,21\}$. List the elements in the set $(D \cup E) \cap F$.
$(D \cup E) \cap F=$
Answer (2)
Find the union of sets D and E: D ∪ E = { 16 , 18 , 19 , 20 , 21 } .
Find the intersection of ( D ∪ E ) and F: ( D ∪ E ) ∩ F = { 18 , 19 , 21 } .
List the elements in the resulting set: 18, 19, 21.
The final answer is { 18 , 19 , 21 } .
Explanation
Understanding the Problem We are given three sets: D = { 16 , 19 , 21 } , E = { 16 , 18 , 19 , 20 } , and F = { 15 , 17 , 18 , 19 , 21 } . We want to find the set ( D ∪ E ) ∩ F . This means we first find the union of D and E , which is the set of all elements in either D or E (or both). Then we find the intersection of that union with F , which is the set of elements that are in both ( D ∪ E ) and F .
Finding the Union of D and E First, let's find D ∪ E . This set contains all elements that are in D or E or both. So, D ∪ E = { 16 , 18 , 19 , 20 , 21 } .
Finding the Intersection of (D union E) and F Next, we need to find ( D ∪ E ) ∩ F . This set contains all elements that are in both ( D ∪ E ) and F . We have D ∪ E = { 16 , 18 , 19 , 20 , 21 } and F = { 15 , 17 , 18 , 19 , 21 } . The elements that are in both of these sets are 18, 19, and 21. Therefore, ( D ∪ E ) ∩ F = { 18 , 19 , 21 } .
Final Answer The elements in the set ( D ∪ E ) ∩ F are 18, 19, and 21.
Examples
Understanding set operations like union and intersection is crucial in many areas, such as database management and data analysis. For instance, imagine you have two lists of customers: one list of customers who purchased product A and another list of customers who purchased product B. The union of these lists would give you all customers who purchased either product A or product B, while the intersection would give you the customers who purchased both products. This helps in targeted marketing and understanding customer behavior.
To find ( D ∪ E ) ∩ F , we first determine the union of sets D and E , which is { 16 , 18 , 19 , 20 , 21 } . Next, we find the intersection with set F , leading to the final answer of { 18 , 19 , 21 } . Therefore, the result is { 18 , 19 , 21 } .
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