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Since U (universal set) contains everything (including itself), naturally its opposite would be the set ∅ (empty set) which contains nothing.

But if U contains everything, would that not include ∅? And if ∅ is a part of U, then can it be its opposite?

Opposites will Venn Diagram with no overlap, but U and ∅ would be a "donut" (albeit a donut with a very small hole that might not exist, but that is beside the point).

So which way does this work? Am I breaking logic? Is it breaking logic? Or is there a rational explanation that I am missing?

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    What is $U$ here? – 2017-01-19
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    The universal set. Sorry, will edit to make that clear. – 2017-01-19
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    The universal set can't exist. – 2017-01-19
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    This might be a dumb question. It can't? – 2017-01-19
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    @positrón0802: There are formulations of set theory with a universal set. Quine's $NF$ is one of the best known and most widelybstudied. See https://en.wikipedia.org/wiki/Universal_set. (And, in $NF$, $U$ includes everything, including $\emptyset$ and itself.) – 2017-01-19

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There are two types of "containment" you could talk about, with the idea that something is in something else.

First, if you have two sets $A$ and $B$, you could have $A \subseteq B$, which we read as "$A$ is contained in $B$". This means every element of $A$ is in $B$. A venn diagram of this would show two concentric circles, with $A$ being the inner circle and $B$ being the outer circle.

We also have $\in$. For example, I can say $x \in A$. This means $A$ is a set, and one of the objects inside of $A$ is $x$. I can also express $\in$ using the containment (subset) notion I described above. Instead of writing $x \in A$, I can write $\{x\} \subseteq A$.

Now, if $U$ is the universal set, that means it's the largest set we are considering in some example. It's the biggest set in that example, and we can talk about new sets only by taking subsets of the universal set. Since $U$ is the universal set, every new set we form has to be a subset of it. Also, yes, $\emptyset$ is contained in $U$. But we mean $\emptyset \subseteq U$. We don't mean $\emptyset \in U$. We could have $\emptyset \in U$ if $U$ happens to be a set of sets and you clearly state/define that $\emptyset \in U$. But otherwise, there's no reason for $\emptyset \in U$ to be true. On the other hand, $\emptyset \subseteq U$ is always true, and this is because of vacuous truth. I recommend you look that up if you are still confused.

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    That may well be the right answer for the OP's context, but do note that when set theorists talk about "set theory with a universal set", they mean set theory with a set $U$ whose elements include everything (including $U$ itself). – 2017-01-19
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    @RobArthan I know. :) Thank you for reminding me and enlightening the community and OP. – 2017-01-19