Let:
[tex]\begin{array}{l}
U=\{b, f, h, m, o, s, t\} \\
X=\{b, h, o, t\} \\
Y=\{b, f, h\} \\
Z=\{f, h, m, o, s\}
\end{array}[/tex]

Find the following set: [tex]\left(Z \cup X^{\prime}\right)^{\prime} \cap Y[/tex]
[tex]\left(Z \cup X^{\prime}\right)^{\prime} \cap Y=\{ \}[/tex]
(Use a comma to separate answers as needed.)

Find the complement of set X : X ′ = { f , m , s } .
Find the union of sets Z and X ′ : Z ∪ X ′ = { f , h , m , o , s } .
Find the complement of the union of Z and X ′ : ( Z ∪ X ′ ) ′ = { b , t } .
Find the intersection of ( Z ∪ X ′ ) ′ and Y : ( Z ∪ X ′ ) ′ ∩ Y = { b } .

{ b } ​
Explanation

Understanding the Problem We are given the following sets:

U = { b , f , h , m , o , s , t } X = { b , h , o , t } Y = { b , f , h } Z = { f , h , m , o , s }
We need to find the set ( Z ∪ X ′ ) ′ ∩ Y .

Finding the Complement of X First, we need to find the complement of X , denoted as X ′ . The complement of X contains all elements in the universal set U that are not in X .

X ′ = U − X = { b , f , h , m , o , s , t } − { b , h , o , t } = { f , m , s }

Finding the Union of Z and X' Next, we need to find the union of Z and X ′ , denoted as Z ∪ X ′ . The union of two sets contains all elements that are in either set.

Z ∪ X ′ = { f , h , m , o , s } ∪ { f , m , s } = { f , h , m , o , s }

Finding the Complement of Z union X' Now, we need to find the complement of Z ∪ X ′ , denoted as ( Z ∪ X ′ ) ′ . This contains all elements in the universal set U that are not in Z ∪ X ′ .

( Z ∪ X ′ ) ′ = U − ( Z ∪ X ′ ) = { b , f , h , m , o , s , t } − { f , h , m , o , s } = { b , t }

Finding the Intersection Finally, we need to find the intersection of ( Z ∪ X ′ ) ′ and Y , denoted as ( Z ∪ X ′ ) ′ ∩ Y . The intersection of two sets contains all elements that are common to both sets.

( Z ∪ X ′ ) ′ ∩ Y = { b , t } ∩ { b , f , h } = { b }

Final Answer Therefore, ( Z ∪ X ′ ) ′ ∩ Y = { b }

Examples
Set theory is used in computer science for data analysis and database management. For example, when querying a database, set operations like union, intersection, and complement are used to retrieve specific data sets based on certain criteria. Understanding set theory helps in optimizing database queries and managing data efficiently. Also, in web development, set theory can be used to manage user permissions and access control, ensuring that users have the appropriate level of access to different parts of a website or application.

Answered by GinnyAnswer | 2025-07-07