Question

if the set u = {all positive integers} and set a = {x|x ∈ u and x is an odd positive integer}, which describes the complement of set a, a^c?
a^c = {x|x ∈ u and is an even positive integer}
a^c = {x|x ∈ u and is not an integer}
a^c = {x|x ∈ u and is zero}
a^c = {x|x ∈ u and is a negative integer}

Explanation:

Step1: Recall Complement Definition

The complement of a set \( A \) (denoted \( A^c \)) in the universal set \( U \) is the set of all elements in \( U \) that are not in \( A \).

Step2: Analyze Set \( U \) and \( A \)

• \( U = \{\text{all positive integers}\} \)
• \( A = \{x | x \in U \text{ and } x \text{ is an odd positive integer}\} \). So, elements in \( A \) are odd positive integers.

Step3: Determine \( A^c \)

Elements in \( U \) not in \( A \) must be positive integers that are not odd. The only other type of positive integer is even. So, \( A^c = \{x | x \in U \text{ and } x \text{ is an even positive integer}\}. \)

Step4: Evaluate Options

• Option 1: Matches our conclusion.
• Option 2: \( U \) is positive integers, so "not an integer" is invalid.
• Option 3: Zero is not a positive integer (since \( U \) is positive integers), so invalid.
• Option 4: Negative integers are not in \( U \) (positive integers), so invalid.

Answer:

\( \text{A}^c = \{x|x \in \cup \text{ and is an even positive integer}\} \) (the first option)