Question

Let S be the sample space of the numbers greater than 0 and less than 12. Let A be a subset of S, which consists of the even numbers. What is the complement of set A?

Answer 1

The sample space \( S \) consists of all numbers greater than 0 and less than 12. Thus, we can express \( S \) as follows:

\[
S = \{ x \in \mathbb{R} \mid 0 < x < 12 \}
\]

The subset \( A \), which consists of the even numbers within this range, includes the even integers between 1 and 12. Therefore, we can specify set \( A \) as:

\[
A = \{ 2, 4, 6, 8, 10 \}
\]

To find the complement of set \( A \) (denoted as \( A' \)), we consider all elements in the sample space \( S \) that are not in \( A \).

Thus, \( A' \) includes all numbers in the interval \( (0, 12) \) except for the even numbers in \( A \). This can be expressed as:

\[
A' = S \setminus A = \{ x \in S \mid x \notin A \}
\]

The odd integers in the same range are:

\[
\{ 1, 3, 5, 7, 9, 11 \}
\]

Additionally, it includes all the non-integer numbers in the interval (0, 12) that are not even.

Therefore, the complement of set \( A \) can be summarized as:

\[
A' = \{ x \in (0, 12) \mid x \text{ is not in } \{ 2, 4, 6, 8, 10 \} \}
\]

This total set would include all odd numbers and all non-integer numbers in the range (0, 12).