Let's carefully determine the complement of the set [tex]\( S \)[/tex] given the problem description.
### Step-by-Step Solution:
1. Define the Set [tex]\( S \)[/tex]:
The set [tex]\( S \)[/tex] is defined as:
[tex]\[
S = \{ x \mid x < 5 \}
\][/tex]
This set includes all real numbers that are less than 5.
2. Understand the Concept of Complement:
The complement of a set [tex]\( S \)[/tex], denoted as [tex]\( S' \)[/tex], is the set of all elements in the universal set that are not in [tex]\( S \)[/tex].
3. Determine the Universal Set:
In this problem, the universal set is all real numbers, typically denoted by [tex]\( \mathbb{R} \)[/tex].
4. Find the Complement [tex]\( S' \)[/tex]:
Since [tex]\( S \)[/tex] includes all [tex]\( x \)[/tex] such that [tex]\( x < 5 \)[/tex], the complement will include all [tex]\( x \)[/tex] that are not less than 5. This can be expressed as:
[tex]\[
S' = \{ x \mid x \geq 5 \}
\][/tex]
5. Check the Given Options:
Let's evaluate the given options for the complement of [tex]\( S \)[/tex]:
- [tex]\(\{ x \mid x \neq 5 \}\)[/tex]: This set includes all real numbers except 5, which is not correct since it excludes numbers like 4.9 or 5.1.
- [tex]\(\{ 6, 7, 8 \ldots \}\)[/tex]: This set includes only natural numbers greater than or equal to 6, which is incorrect. It does not account for numbers like 5, 5.5, etc.
- [tex]\(\{ x \mid x > 5 \}\)[/tex]: This set includes all real numbers greater than 5. This misses out on the number 5 itself.
- [tex]\(\{ x \mid x \geq 5 \}\)[/tex]: This set includes all real numbers greater than or equal to 5, which is correct as it includes the number 5 and all numbers greater than 5.
6. Conclusion:
Therefore, the complement of [tex]\( S \)[/tex] is:
[tex]\[
\boxed{\{ x \mid x \geq 5 \}}
\][/tex]
