To determine the complement of the set [tex]\( S \)[/tex], we need to find all the elements in the universal set that are not in [tex]\( S \)[/tex]. Given that the universal set includes all real numbers and the set [tex]\( S \)[/tex] is defined as:
[tex]\[ S = \{x \mid x < 5\} \][/tex]
We are looking for the set of all [tex]\( x \)[/tex] that do not satisfy this condition.
Step-by-step solution to find the complement of [tex]\( S \)[/tex]:
1. Understand the definition of [tex]\( S \)[/tex]:
The set [tex]\( S \)[/tex] includes all real numbers less than 5. In other words, if [tex]\( x \)[/tex] is an element of [tex]\( S \)[/tex], then [tex]\( x < 5 \)[/tex].
2. Define the complement of [tex]\( S \)[/tex]:
The complement of [tex]\( S \)[/tex], denoted by [tex]\( S' \)[/tex], includes all real numbers that are not in [tex]\( S \)[/tex]. This means any real number [tex]\( x \)[/tex] that does not satisfy [tex]\( x < 5 \)[/tex].
3. Formulate the condition:
To find [tex]\( S' \)[/tex], we need to take all elements [tex]\( x \)[/tex] in the universal set of real numbers and exclude the elements that are in [tex]\( S \)[/tex]. This means all [tex]\( x \)[/tex] such that [tex]\( x \geq 5 \)[/tex].
Therefore, the complement of [tex]\( S \)[/tex] is given by:
[tex]\[ S' = \{x \mid x \geq 5\} \][/tex]
Review the options provided:
- [tex]\( \{x \mid x \neq 5\} \)[/tex]: This set includes all real numbers except 5. This is not the complement of [tex]\( S \)[/tex] since it includes numbers less than 5 and greater than 5, excluding only 5.
- [tex]\( \{6, 7, 8, \ldots\} \)[/tex]: This set includes only integers greater than 5 and not all real numbers greater than or equal to 5, so it's incorrect.
- [tex]\( \{x \mid x > 5\} \)[/tex]: This set includes all real numbers greater than 5 but excludes 5 itself. Therefore, this is not the correct complement set.
- [tex]\( \{x \mid x \geq 5\} \)[/tex]: This set includes all real numbers greater than or equal to 5, which correctly represents the complement of [tex]\( S \)[/tex].
The correct answer is:
[tex]\[ \{x \mid x \geq 5\} \][/tex]
Thus, we conclude that the complement of [tex]\( S \)[/tex] is:
[tex]\[ \{x \mid x \geq 5\} \][/tex]
In terms of the provided options, the correct one is the fourth option.
