1) Given the universal set
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and sets A = {2, 3, 4, 6},
B = {5, 6, 7, 8, 10},
C = {6, 8, 10}, and D = {1, 2, 7, 9}, answering the following:
1- a) What is A ∪ C?
2- b) What is A ∩ D?
3- c) What is A'∩ (B ∪ C)?
4- d) What is A ∪ B ∪ C ∪ D?
5- e) What is (A ∩ B') ∪ (C ∩ D')?
To answer these questions, we need to understand the basic principles of set theory and how to perform operations on sets. Here's a step-by-step explanation of how to find the answers:
1) a) What is A ∪ C?
To find the union of sets A and C (A ∪ C), we need to combine all the elements that are in either A or C, or both. In this case, A = {2, 3, 4, 6} and C = {6, 8, 10}. The union of A and C will be {2, 3, 4, 6, 8, 10}.
2) b) What is A ∩ D?
To find the intersection of sets A and D (A ∩ D), we need to identify elements that are common to both sets. In this case, A = {2, 3, 4, 6} and D = {1, 2, 7, 9}. The intersection of A and D will be {2}.
3) c) What is A'∩(B ∪ C)?
To find the complement of set A (A') and the intersection of the complement of A with the union of B and C, we perform the following steps:
- First, we find the complement of set A. The complement of A (A') will consist of all the elements in the universal set U that are not in A. In this case, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 3, 4, 6}. Therefore, A' = {1, 5, 7, 8, 9, 10}.
- Next, we find the union of sets B and C (B ∪ C). This is the combination of all elements in either B or C, or both. In this case, B = {5, 6, 7, 8, 10} and C = {6, 8, 10}. Therefore, B ∪ C = {5, 6, 7, 8, 10}.
- Finally, we find the intersection of A' and (B ∪ C). The intersection of two sets is the set of elements that are common to both sets. Applying this to A' and (B ∪ C), we get A' ∩ (B ∪ C) = {5, 7, 8, 10}.
4) d) What is A ∪ B ∪ C ∪ D?
To find the union of sets A, B, C, and D (A ∪ B ∪ C ∪ D), we combine all the elements in these sets. In this case, A = {2, 3, 4, 6}, B = {5, 6, 7, 8, 10}, C = {6, 8, 10}, and D = {1, 2, 7, 9}. The union of A, B, C, and D will be {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (which is the universal set, U).
5) e) What is (A ∩ B') ∪ (C ∩ D')?
To find the union of (A ∩ B') and (C ∩ D'), we first need to find (A ∩ B') and (C ∩ D') separately:
- (A ∩ B') represents the intersection of set A with the complement of set B. First, we find the complement of B. B' (B complement) consists of all the elements in the universal set U that are not in B. In this case, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {5, 6, 7, 8, 10}. Therefore, B' = {1, 2, 3, 4, 9}. Next, we find the intersection of A with B', which will be {2, 3, 4, 6} (since B' does not have element 6).
- (C ∩ D') represents the intersection of set C with the complement of set D. First, we find the complement of D. D' (D complement) consists of all the elements in the universal set U that are not in D. In this case, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and D = {1, 2, 7, 9}. Therefore, D' = {3, 4, 5, 6, 8, 10}. Next, we find the intersection of C with D', which will be {6, 8, 10} (since D' does not have elements 1, 2, 7, or 9).
- Finally, we find the union of (A ∩ B') and (C ∩ D'). The union of two sets is the combination of all elements in either set. In this case, (A ∩ B') ∪ (C ∩ D') = {2, 3, 4, 6, 8, 10}.
I have explained how to find the answers to the given questions using set theory principles. If you have any further questions, feel free to ask!