The universal set, [tex]$U$[/tex], contains 60 elements. If [tex]$X \subset U$[/tex] and [tex]$n(X)=21$[/tex], then what is the value of [tex]$n(X')$[/tex]?
Answer (2)
The universal set U has 60 elements, and set X has 21 elements.
The number of elements in the complement of X is found by subtracting the number of elements in X from the number of elements in U .
Calculate n ( X ′ ) = n ( U ) − n ( X ) = 60 − 21 .
The number of elements in the complement of X is 39 .
Explanation
Understand the problem and provided data We are given a universal set U with 60 elements, so n ( U ) = 60 . We also have a subset X of U with 21 elements, so n ( X ) = 21 . Our goal is to find the number of elements in the complement of X , denoted as n ( X ′ ) .
Recall the formula for the complement of a set The complement of a set X , denoted as X ′ , contains all the elements in the universal set U that are not in X . The formula for finding the number of elements in the complement is: n ( X ′ ) = n ( U ) − n ( X ) This formula tells us that to find the number of elements in X ′ , we subtract the number of elements in X from the total number of elements in the universal set U .
Substitute the values and calculate Now, we substitute the given values into the formula: n ( X ′ ) = 60 − 21 Performing the subtraction, we get: n ( X ′ ) = 39 So, the number of elements in the complement of X is 39.
State the final answer Therefore, the value of n ( X ′ ) is 39.
Examples
Imagine a class of 60 students where 21 students are part of the math club. To find out how many students are NOT in the math club, you would calculate the complement. This is a practical application of set theory in everyday situations.
The number of elements in the complement of set X is calculated using the formula n ( X ′ ) = n ( U ) − n ( X ) . Substituting the values gives n ( X ′ ) = 60 − 21 = 39 . Therefore, the value of n ( X ′ ) is 39.
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