If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 3, 4,} B = {2, 4, 6, 8), then (A ∪ B)' is -
Understanding Set Operations
In set theory, we work with collections of distinct objects called sets. The question asks us to find the complement of the union of two sets, \( A \) and \( B \), with respect to a universal set \( U \).
We are given the following sets:
- Universal Set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- Set \( A = \{1, 2, 3, 4\} \)
- Set \( B = \{2, 4, 6, 8\} \)
Calculating the Union of Sets A and B
The union of two sets, denoted by \( A \cup B \), is the set containing all the elements that are in set \( A \) or in set \( B \) or in both.
To find \( A \cup B \), we combine the elements from set \( A \) and set \( B \), listing each element only once.
\( A = \{1, 2, 3, 4\} \)
\( B = \{2, 4, 6, 8\} \)
\( A \cup B = \{1, 2, 3, 4\} \cup \{2, 4, 6, 8\} \)
The elements in \( A \) are 1, 2, 3, 4. The elements in \( B \) are 2, 4, 6, 8. Combining these unique elements gives us:
\( A \cup B = \{1, 2, 3, 4, 6, 8\} \)
Calculating the Complement of the Union
The complement of a set \( S \), denoted by \( S' \) or \( S^c \), with respect to a universal set \( U \), is the set of all elements in \( U \) that are not in \( S \). In this problem, we need to find \( (A \cup B)' \). This means we need to find all elements in the universal set \( U \) that are not in the set \( A \cup B \).
We have:
- Universal Set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- Set \( A \cup B = \{1, 2, 3, 4, 6, 8\} \)
To find \( (A \cup B)' \), we look at the elements in \( U \) and remove the elements that are also in \( A \cup B \).
Elements in \( U \) are: 1, 2, 3, 4, 5, 6, 7, 8, 9.
Elements in \( A \cup B \) are: 1, 2, 3, 4, 6, 8.
Removing the elements {1, 2, 3, 4, 6, 8} from {1, 2, 3, 4, 5, 6, 7, 8, 9}, we are left with:
\( (A \cup B)' = \{5, 7, 9\} \)
Final Result
The complement of the union of sets A and B, \( (A \cup B)' \), is the set containing the elements 5, 7, and 9.
Important Questions from Operations on Sets
Match List I with List II
Let R 1= {(1, 1), (2, 2), (3, 3)} and R 2 = {(1, 1), (1, 2), (1, 3), (1, 4)}
List I
List II
(A) R 1∪ R 2
(I) {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (3, 3)}
(B) R 1- R 2
(II) {1, 1}
(C) R 1∩ R 2
(III) {(1, 2), (1, 3), (1, 4)}
(D) R 2- R 1
(IV) {(2, 2), (3, 3)}
Choose the correct answer from the options given below:
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