Given that the universal set is all real numbers and set S is defined as the set of all [tex]$x$[/tex] such that [tex]$x$[/tex] is less than 5, which of the following is the complement of S?
[tex]S={x \mid x\ \textless \ 5}[/tex]
A. [tex]{x \mid x \neq 5}[/tex]
B. [tex]{6,7,8 \ldots}[/tex]
C. [tex]{x \mid x\ \textgreater \ 5}[/tex]
D. [tex]{x \mid x \geq 5}[/tex]
Answer (2)
The universal set is all real numbers, and S = { x ∣ x < 5 } .
The complement of S, S c , includes all elements in the universal set that are not in S.
S c contains all real numbers greater than or equal to 5.
Therefore, S c = { x ∣ x ≥ 5 } , so the answer is { x ∣ x ≥ 5 } .
Explanation
Understanding the Problem The universal set is the set of all real numbers, and the set S is defined as all x such that x < 5 . We want to find the complement of S, which includes all elements in the universal set that are not in S.
Defining the Complement The complement of a set S, denoted as S c , consists of all elements in the universal set that are not in S. In this case, the universal set is all real numbers, so S c will contain all real numbers that are not less than 5.
Determining the Complement This means S c will contain all real numbers greater than or equal to 5. In set notation, this is represented as S c = { x ∣ x ≥ 5 } .
Final Answer Therefore, the complement of S is the set of all x such that x is greater than or equal to 5.
Examples
Imagine you're organizing a race where only participants under 5 years old are allowed in the 'Under 5' category. The complement represents everyone else—those who are 5 years old or older. Understanding complements helps in scenarios like filtering data, defining eligibility criteria, or categorizing items based on specific conditions. This concept is useful in computer science for data analysis, in statistics for defining event spaces, and in everyday life for making decisions based on criteria.
The complement of the set S, which includes all real numbers less than 5, is the set of all real numbers greater than or equal to 5. Therefore, the correct choice is D: { x ∣ x ≥ 5 } . This set includes all numbers starting from 5 and going on infinitely upward.
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