Use the sets to find $F \cap(D \cup E)$. Let $D=\{12,15,17\}, E=\{12,14,15,16\}$, and $F=\{11,13,14,15,17\}$. $F \cap(D \cup E)=$ $\square$ (Use a comma to separate answers as needed.)
Answer (1)
First, find the union of sets D and E : D ∪ E = { 12 , 14 , 15 , 16 , 17 } .
Next, find the intersection of set F with the union D ∪ E : F ∩ ( D ∪ E ) = { 14 , 15 , 17 } .
The final answer is { 14 , 15 , 17 } .
Explanation
Understanding the Problem We are given three sets: D = { 12 , 15 , 17 } , E = { 12 , 14 , 15 , 16 } , and F = { 11 , 13 , 14 , 15 , 17 } . Our goal is to find the intersection of set F with the union of sets D and E , which is written as F ∩ ( D ∪ E ) .
Finding the Union of D and E First, we need to find the union of sets D and E . The union of two sets contains all the elements that are in either set or in both. So, D ∪ E = { 12 , 15 , 17 } ∪ { 12 , 14 , 15 , 16 } = { 12 , 14 , 15 , 16 , 17 } .
Finding the Intersection of F and (D union E) Next, we need to find the intersection of set F with the union we just found, D ∪ E . The intersection of two sets contains all the elements that are in both sets. So, F ∩ ( D ∪ E ) = { 11 , 13 , 14 , 15 , 17 } ∩ { 12 , 14 , 15 , 16 , 17 } = { 14 , 15 , 17 } .
Final Answer Therefore, F ∩ ( D ∪ E ) = { 14 , 15 , 17 } .
Examples
Understanding set operations like union and intersection is crucial in many real-world scenarios. For instance, imagine you're organizing a party and need to send out invitations. Set D represents people who like dancing, and set E represents people who like eating. The union of D and E (D ∪ E) would be the list of all people who like either dancing or eating or both. Now, if set F represents your list of friends, then F ∩ (D ∪ E) would give you the list of friends who enjoy either dancing or eating, ensuring you invite the right people to your party. This concept extends to database queries, data analysis, and even network design, where understanding relationships between different groups or categories is essential.
