Given U = {1, 2, 4, 5, 7, 9}, B = {1, 2, 4, 7}, and D = {1, 4, 9}: Find (a) B ∪ D' and (b) B' ∩ D'.
Understand the Problem
The question involves set theory and asks us to find the union and intersection of sets. First, we need to determine the complement of set D (D'). Then, for part (a), we find the union of set B and D'. For part (b), we need to find the complement of set B (B') and determine the intersection of B' and D'.
Answer
(a) $\{1, 2, 4, 5, 7\}$ (b) $\{5\}$
Answer for screen readers
(a) $B \cup D' = {1, 2, 4, 5, 7}$ (b) $B' \cap D' = {5}$
Steps to Solve
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Find the complement of set D (D') The complement of a set D, denoted by D', is the set of all elements in the universal set U that are not in D. $U = {1, 2, 4, 5, 7, 9}$ $D = {1, 4, 9}$ $D' = U - D = {2, 5, 7}$
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Find the union of B and D' The union of two sets B and D', denoted by $B \cup D'$, is the set of all elements that are in B or in D' or in both. $B = {1, 2, 4, 7}$ $D' = {2, 5, 7}$ $B \cup D' = {1, 2, 4, 5, 7}$
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Find the complement of set B (B') The complement of a set B, denoted by B', is the set of all elements in the universal set U that are not in B. $U = {1, 2, 4, 5, 7, 9}$ $B = {1, 2, 4, 7}$ $B' = U - B = {5, 9}$
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Find the intersection of B' and D' The intersection of two sets B' and D', denoted by $B' \cap D'$, is the set of all elements that are in both B' and D'. $B' = {5, 9}$ $D' = {2, 5, 7}$ $B' \cap D' = {5}$
(a) $B \cup D' = {1, 2, 4, 5, 7}$ (b) $B' \cap D' = {5}$
More Information
The union of two sets combines all unique elements from both sets. The intersection of two sets contains only the elements that are common to both sets. The complement of a set contains elements from the universal set that are not in the original set.
Tips
A common mistake is to include duplicate elements when finding the union of two sets. Remember to only list each unique element once. Another mistake is to confuse union and intersection. Union includes all elements from both sets, while intersection only includes elements present in both sets. Also, students may have trouble with complements, so it's important to carefully remove the elements of the original set from the universal set.
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