For the universal set, $U=\{2,3,4,5,6,7\}$, complete the parts below. Write your answers in roster form or as $\varnothing$.
(a) Suppose $A=\{4,5,7\}$. Then what is $A^{\prime}$ ?
$A^{\prime}=\square$
(b) Suppose we know that $C^{\prime}=\{2,5,6,7\}$. Then what would $C$ have to be?
$C=$
$\square$
Answer (1)
Find A ′ by identifying elements in U that are not in A : A ′ = { 2 , 3 , 6 } .
Find C by identifying elements in U that are not in C ′ : C = { 3 , 4 } .
The complement of A is A ′ = { 2 , 3 , 6 } .
The set C is C = { 3 , 4 } .
Explanation
Understanding the Problem We are given the universal set U = { 2 , 3 , 4 , 5 , 6 , 7 } . We need to find the complement of set A and the set C given their complements with respect to U .
Finding A' (a) We are given A = { 4 , 5 , 7 } . The complement of A , denoted as A ′ , consists of all elements in U that are not in A . So, we look for elements in U that are not in A .
Listing elements of A' The elements in U are 2, 3, 4, 5, 6, and 7. The elements in A are 4, 5, and 7. Therefore, the elements in A ′ are 2, 3, and 6. Thus, A ′ = { 2 , 3 , 6 } .
Finding C (b) We are given C ′ = { 2 , 5 , 6 , 7 } . The set C consists of all elements in U that are not in C ′ .
Listing elements of C The elements in U are 2, 3, 4, 5, 6, and 7. The elements in C ′ are 2, 5, 6, and 7. Therefore, the elements in C are 3 and 4. Thus, C = { 3 , 4 } .
Final Answer Therefore, A ′ = { 2 , 3 , 6 } and C = { 3 , 4 } .
Examples
Understanding set complements is crucial in database management. For instance, if you have a database of all customers (the universal set) and a subset of customers who purchased a specific product (set A), finding the complement A' helps identify customers who have not purchased that product. This information can be used for targeted marketing campaigns to encourage those customers to buy the product. Similarly, if you know the set of users who haven't accessed a feature (C'), you can find the set of users who have (C) to gather feedback or improve user experience.
