If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {1, 2, 4, 6, 7}, then find:
1) A ∪ B
2) B ∪ C
3) (A ∪ B) ∪ C
4) A ∩ (B ∪ C)
5) A ∩ (B ∩ C)
6) A ∪ (B ∪ C)
What is your observation?
Answer (2)
The sets' union operations result in combinations of unique elements, while the intersection results focus on shared elements. Specifically, A ∪ B = {1, 2, 3, 4, 5, 6} and A ∩ (B ∩ C) = {4}. The findings highlight the distinct characteristics of union and intersection in set theory.
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To solve this problem, we'll use set operations to find the union and intersection between the given sets A, B, and C. Let's analyze each part step by step:
A \cup B : \ The union of sets A and B includes all the elements that are in either set A, set B, or both. \ A = { 1 , 2 , 3 , 4 } \ B = { 3 , 4 , 5 , 6 } \ Union: \ A ∪ B = { 1 , 2 , 3 , 4 , 5 , 6 } \
B \cup C : \ The union of sets B and C includes all elements that are in either set B, set C, or both. \ C = { 1 , 2 , 4 , 6 , 7 } \ Union: \ B ∪ C = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } \
(A \cup B) \cup C : \ First, find A ∪ B , which we previously found as { 1 , 2 , 3 , 4 , 5 , 6 } . Then, take the union with set C. \ ( A ∪ B ) ∪ C = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } \
A \cap (B \cup C) : \ First, find B ∪ C = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . Then, find the intersection with set A. This includes the elements that are in both A and B ∪ C . \ A ∩ ( B ∪ C ) = { 1 , 2 , 4 } \
A \cap (B \cap C) : \ First, find the intersection of sets B and C: \ B ∩ C = { 4 , 6 } \ Then, find the intersection with set A: \ A ∩ ( B ∩ C ) = { 4 } \
A \cup (B \cup C) : \ First, find B ∪ C = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . Then, take the union with set A. \ A ∪ ( B ∪ C ) = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } \
Observation:
The union operation combines all elements from the sets, while eliminating duplicates.
The intersection operation finds common elements between the sets.
The solutions illustrate the relationship and interaction between multiple sets, such as how elements can be distributed among different groups.
