If the set $U=\{$ all positive integers $\}$ and set $A=\{x \mid x \in U$ and $x$ is an odd positive integer $\}$, which describes the complement of set $A, A^c$ ?
A. $A^c=\{x \mid x \in U$ and is a negative integer $\}$
B. $A^c=\{x \mid x \in U$ and is zero $\}$
C. $A^c=\{x \mid x \in U$ and is not an integer $\}$
D. $A^c=\{x \mid x \in U$ and is an even positive integer $\}$
Answer (1)
The universal set U is all positive integers, and set A is all odd positive integers.
The complement of A , denoted A c , contains elements in U but not in A .
Therefore, A c consists of all positive integers that are not odd, which are the even positive integers.
The complement of set A is the set of all even positive integers: A c = { x ∣ x ∈ U and x is an even positive integer } .
Explanation
Understanding the Problem We are given the universal set U as the set of all positive integers, and the set A as the set of all odd positive integers. We need to find the complement of set A , denoted as A c . The complement of a set A contains all elements in the universal set U that are not in A .
Identifying the Complement Since U consists of all positive integers and A consists of all odd positive integers, the complement A c will consist of all positive integers that are not odd.
Determining the Elements of the Complement The positive integers that are not odd are the even positive integers. Therefore, A c is the set of all even positive integers.
Final Answer The complement of set A , A c , is the set of all even positive integers.
Examples
In a classroom of students, if the universal set U is all students in the class, and set A is all the students who are wearing glasses, then the complement of set A , A c , would be the set of all students who are not wearing glasses. This concept is useful in many areas, such as data analysis, where you might want to identify the subset of data that does not meet a certain criterion.
