Let the universal set be the set [tex]$R$[/tex] of all real numbers and 1. [tex]$A=\{x \in R \mid-3 \leq x \leq 0\}, B=\{x \in R \mid-1\ \textless \ x\ \textless \ 2\}$[/tex], and [tex]$C=\{x \in R \mid 6\ \textless \ x \leq 8\}$[/tex]. Find each of the following:
a. [tex]$A \cup B$[/tex]
b. [tex]$A \cap B$[/tex]
c. [tex]$A^c$[/tex]
d. [tex]$A \cup C$[/tex]
e. [tex]$A \cap C$[/tex]
f. [tex]$B^c$[/tex]
g. [tex]$A^c \cap B^c$[/tex]
h. [tex]$A^c \cup B^c$[/tex]
i. [tex]$(A \cap B)^c$[/tex]
j. [tex]$(A \cup B)^c$[/tex]
Answer (2)
We computed various operations on the sets A, B, and C, including unions, intersections, and complements. The results show how these sets relate in terms of shared and distinct elements. These operations are fundamental in understanding set theory in mathematics.
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Find the union of A and B: A ∪ B = [ − 3 , 2 )
Find the intersection of A and B: A ∩ B = ( − 1 , 0 ]
Find the complement of A: A c = ( − ∞ , − 3 ) ∪ ( 0 , ∞ )
Find the complement of B: B c = ( − ∞ , − 1 ] ∪ [ 2 , ∞ )
Use De Morgan's Laws to find the intersection and union of complements.
The final answers are: a. A ∪ B = [ − 3 , 2 ) b. A ∩ B = ( − 1 , 0 ] c. A c = ( − ∞ , − 3 ) ∪ ( 0 , ∞ ) d. A ∪ C = [ − 3 , 0 ] ∪ ( 6 , 8 ] e. A ∩ C = ∅ f. B c = ( − ∞ , − 1 ] ∪ [ 2 , ∞ ) g. A c ∩ B c = ( − ∞ , − 3 ) ∪ [ 2 , ∞ ) h. A c ∪ B c = ( − ∞ , − 1 ] ∪ ( 0 , ∞ ) i. ( A ∩ B ) c = ( − ∞ , − 1 ] ∪ ( 0 , ∞ ) j. ( A ∪ B ) c = ( − ∞ , − 3 ) ∪ [ 2 , ∞ )
Explanation
Understanding the Sets We are given three sets A , B , and C that are subsets of the real numbers R . We need to find the union, intersection, and complement of these sets. Let's first define the sets in interval notation:
A = [ − 3 , 0 ] B = ( − 1 , 2 ) C = ( 6 , 8 ]
Finding A union B a. A ∪ B : This is the union of sets A and B , which includes all elements in either A or B or both. Since A = [ − 3 , 0 ] and B = ( − 1 , 2 ) , we have
A ∪ B = [ − 3 , 2 )
Finding A intersection B b. A ∩ B : This is the intersection of sets A and B , which includes all elements that are in both A and B . Since A = [ − 3 , 0 ] and B = ( − 1 , 2 ) , we have
A ∩ B = ( − 1 , 0 ]
Finding the complement of A c. A c : This is the complement of set A , which includes all elements in R that are not in A . Since A = [ − 3 , 0 ] , we have
A c = ( − ∞ , − 3 ) ∪ ( 0 , ∞ )
Finding A union C d. A ∪ C : This is the union of sets A and C , which includes all elements in either A or C or both. Since A = [ − 3 , 0 ] and C = ( 6 , 8 ] , we have
A ∪ C = [ − 3 , 0 ] ∪ ( 6 , 8 ]
Finding A intersection C e. A ∩ C : This is the intersection of sets A and C , which includes all elements that are in both A and C . Since A = [ − 3 , 0 ] and C = ( 6 , 8 ] , there are no elements in common, so
A ∩ C = ∅
Finding the complement of B f. B c : This is the complement of set B , which includes all elements in R that are not in B . Since B = ( − 1 , 2 ) , we have
B c = ( − ∞ , − 1 ] ∪ [ 2 , ∞ )
Finding the intersection of the complements of A and B g. A c ∩ B c : This is the intersection of the complements of A and B . Using De Morgan's Law, A c ∩ B c = ( A ∪ B ) c . From part a, A ∪ B = [ − 3 , 2 ) , so
A c ∩ B c = ( − ∞ , − 3 ) ∪ [ 2 , ∞ )
Finding the union of the complements of A and B h. A c ∪ B c : This is the union of the complements of A and B . Using De Morgan's Law, A c ∪ B c = ( A ∩ B ) c . From part b, A ∩ B = ( − 1 , 0 ] , so
A c ∪ B c = ( − ∞ , − 1 ] ∪ ( 0 , ∞ )
Finding the complement of A intersection B i. ( A ∩ B ) c : This is the complement of the intersection of A and B . From part b, A ∩ B = ( − 1 , 0 ] , so
( A ∩ B ) c = ( − ∞ , − 1 ] ∪ ( 0 , ∞ )
Finding the complement of A union B j. ( A ∪ B ) c : This is the complement of the union of A and B . From part a, A ∪ B = [ − 3 , 2 ) , so
( A ∪ B ) c = ( − ∞ , − 3 ) ∪ [ 2 , ∞ )
Final Answer In summary: a. A ∪ B = [ − 3 , 2 ) b. A ∩ B = ( − 1 , 0 ] c. A c = ( − ∞ , − 3 ) ∪ ( 0 , ∞ ) d. A ∪ C = [ − 3 , 0 ] ∪ ( 6 , 8 ] e. A ∩ C = ∅ f. B c = ( − ∞ , − 1 ] ∪ [ 2 , ∞ ) g. A c ∩ B c = ( − ∞ , − 3 ) ∪ [ 2 , ∞ ) h. A c ∪ B c = ( − ∞ , − 1 ] ∪ ( 0 , ∞ ) i. ( A ∩ B ) c = ( − ∞ , − 1 ] ∪ ( 0 , ∞ ) j. ( A ∪ B ) c = ( − ∞ , − 3 ) ∪ [ 2 , ∞ )
Examples
Understanding set operations like union, intersection, and complement is crucial in various real-life scenarios. For instance, consider a marketing campaign where set A represents customers interested in product A and set B represents customers interested in product B. The union (A ∪ B) would be the set of all customers interested in either product, allowing for a broader marketing strategy. The intersection (A ∩ B) would be the set of customers interested in both products, enabling targeted marketing. The complement (Ac) would be the set of customers not interested in product A, helping to refine the marketing focus. These operations help businesses efficiently target their resources and understand their customer base.
