The question ask me to state True or False and give reasons.

However I prefer True.

Reason: The Universal set denoted by (U) is simply a set of all given sets and complement is simply saying that everything that is not in a set. so to say complement of the universal set is just saying everything that is not in the universal set, as result it is an empty set.

I'm I correct?


Yes, you are correct. The universal set $U$ is characterized by $\forall x \colon x \in U$. Taking the complement yields a set $U^{c}$ that is characterized by $\forall x \colon x \not \in U^{c}$. This is equivalent to the statement $\neg \exists x \colon x \in U^{c}$ and hence $U^{c}$ is an (the) empty set. (Depending on your theory, there may not be a unique empty set.)


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