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, Ac?
Ac = {x|x ∈ U and is a negative integer}
Ac = {x|x ∈ U and is zero}
Ac = {x|x ∈ U and is not an integer}
Ac = {x|x ∈ U and is an even positive integer}

Answer 1

Answer: Choice D) set of even integers

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Explanation:

U = universal set = {all positive integers} = {1, 2, 3, 4, 5, ...}

A = {stuff in set U that is odd} = {1, 3, 5, 7, 9, ...}

$A^c$ = {stuff in set U but NOT from set A} = {2, 4, 6, 8, ...}

$A^c$ = {set of even numbers from set U}

In set theory, the complement is simply the complete opposite. If the number 7 is found in set A for instance, then 7 is not going to live in the complement $A^c$

The opposite of the odd numbers is the set of even numbers.

Therefore, we go for choice D as our final answer.

Side note: if we union set A with its complement $A^c$ then we get the universal set as a result. $A \cup A^c = \text{universal set} = \text{set of all positive integers}$