To address the question of whether the statement "If [tex]\( U \)[/tex] is the universal set, then [tex]\( U' = \varnothing \)[/tex]" is true or false, let's break down the concepts involved:
1. Universal Set (U): The universal set [tex]\( U \)[/tex] is a set that contains all possible elements under consideration. It's the "largest" set in a given context, encompassing every element.
2. Complement of the Universal Set ([tex]\( U' \)[/tex]): The complement of the universal set [tex]\( U' \)[/tex] is defined as the set of all elements that are not in [tex]\( U \)[/tex].
To determine if the statement "If [tex]\( U \)[/tex] is the universal set, then [tex]\( U' = \varnothing \)[/tex]" is true, follow these logical steps:
1. Elements of [tex]\( U \)[/tex]: Since [tex]\( U \)[/tex] contains all possible elements, there cannot be any element that exists outside of [tex]\( U \)[/tex].
2. Defining [tex]\( U' \)[/tex]: The complement [tex]\( U' \)[/tex] consists of all elements that are not in [tex]\( U \)[/tex]. Given that [tex]\( U \)[/tex] includes every conceivable element, there are no elements left that could be outside of [tex]\( U \)[/tex].
3. Conclusion: Since no elements remain outside of [tex]\( U \)[/tex], the set [tex]\( U' \)[/tex] must be empty. The absence of any elements outside [tex]\( U \)[/tex] means [tex]\( U' = \varnothing \)[/tex].
Therefore, the statement "If [tex]\( U \)[/tex] is the universal set, then [tex]\( U' = \varnothing \)[/tex]" is true.
So the correct answer is:
A. The statement is true, because [tex]\( U' \)[/tex] means the set of all elements that are not in [tex]\( U \)[/tex], and since [tex]\( U \)[/tex] contains all possible elements, [tex]\( U' \)[/tex] must be the empty set [tex]\(\varnothing\)[/tex].
