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 a negative integer}
a^c = {x|x ∈ u and is zero}
a^c = {x|x ∈ u and is not an integer}
a^c = {x|x ∈ u and is an even positive integer}

Explanation:

The universal set \( U \) is all positive integers. Set \( A \) contains all odd positive integers. The complement of a set \( A \) (denoted \( A^c \)) consists of all elements in \( U \) that are not in \( A \). Since positive integers are either odd or even, the elements not in \( A \) (which has odd positive integers) must be even positive integers.

• The first option is incorrect because \( U \) is positive integers, so negative integers are not in \( U \).
• The second option is incorrect because zero is not a positive integer, so it is not in \( U \).
• The third option is incorrect because all elements of \( U \) are integers, so there are no elements in \( U \) that are not integers.
• The fourth option is correct as it includes all even positive integers (the elements of \( U \) not in \( A \)).

Answer:

D. \( A^c = \{x|x \in U \text{ and is an even positive integer}\} \)