Can sets contain objects of different types? Working on some basic proof work. The conjecture is 
There exists a set $\mathrm X$ for which $\mathbb R \subseteq \mathrm X$ and $\emptyset \in \mathrm X$.
My reasoning was that this is false because the members of $\mathbb R$ are numbers, and $\emptyset$ is a set, hence such a set $\mathrm X$ does not exist.
The book gives $\mathrm X = \mathbb R \cup \{ \emptyset \}$ as a set which satisfies the conditions given in the conjecture.
Any words on where my reasoning is flawed? I was under the impression that sets may only contain elements of the same type. 
 A: The usual set theory (as used as a foundation of mathematics since the early 1900s) is untyped -- everything is just a set, and a real number such as $42$ or $\pi$ will be represented as sets of certain particular shapes. So there's nothing that prevents $\mathbb R\cup\{\varnothing\}$ from existing and being a set.
Actual everyday mathematics outside set theory does use types (in a sorta informal kind of way), so it is very rare that one has use for such mixed-type strange sets in practice. If they appear in an ordinary mathematical argument it is usually a sign that the one who constructed it has not thought thing through properly, or is trying to be way too smart for his own good. But they are not formally forbidden.
One reason to avoid this is that we don't normally want to care exactly how the real numbers get represented as sets. Depending on this choice it may be that $\varnothing$ happens to be the set that represents one of the real numbers -- there are at least arguable technical reasons to choose $\varnothing$ as the set that represents the number zero. In that case $\mathbb R\cup\{\varnothing\}$ will be the same set as $\mathbb R$ itself, probably causing havoc with whatever argument one were trying to use $\mathbb R\cup\{\varnothing\}$ since it won't be true in that case that $(\mathbb R\cup\{\varnothing\})\setminus\mathbb R$ is $\{\varnothing\}$, for example. And then you'll have to prefix all of your theorems with "Assume $\varnothing\notin\mathbb R$" for no practical gain.
There are various attempts to define typed set theories that would align better with how sets are actually used in everyday mathematics, but they have not caught on to the same degree as the untyped ZFC set theory.
A: You can combine objects of any sort into a set. 
There’s nothing wrong with the set $\Bbb R\cup\{\varnothing\}$: its elements are precisely the individual real numbers and the empty set. Another possible choice for $X$ is $X=\Bbb R\cup\wp(\Bbb R)$: the elements of this set are the individual real numbers and the individual sets of real numbers. $\varnothing$ is a set of real numbers, so $\varnothing\in\wp(\Bbb R)$, and therefore $\varnothing\in X$. Similarly, if $x$ is any real number, then $x\in\Bbb R$, and therefore $x\in X$, since $\Bbb R\subseteq X$.
Similarly, there’s nothing wrong with the set
$$A=\Big\{\pi,\sqrt2,\big\{47,\{666\}\big\},\Bbb N\times\Bbb N\Big\}$$
(apart from being a bit hard to read). It has four members: the real number $\pi$, the real number $\sqrt2$, the set $\big\{47,\{666\}\big\}$, and the set $\Bbb N\times\Bbb N$. The set $\big\{47,\{666\}\big\}$ itself has two members, the number $47$ and the set $\{666\}$ whose only element is the number $666$. Note that $47$ is not a member of $A$: it’s a member of a member of $A$. And $666$ isn’t even that: it’s a member of a member of a member of $A$. Finally, the set $\Bbb N\times\Bbb N$, which is a single member of $A$, is itself a moderately complicated set: its members are the ordered pairs of natural numbers.
One more. Let $B=\{a,b\}$, and let $X=B\cup\wp(B)$. We know that
$$\wp(B)=\big\{\varnothing,\{a\},\{b\},\{a,b\}\big\}\;,$$
the set whose members are the sets $\varnothing$, $\{a\}$, $\{b\}$, and $B$ itself, and the members of $B$ are $a$ and $b$, so the members of $B\cup\wp(B)$ are $a,b,\varnothing,\{a\},\{b\}$, and $B$:
$$B\cup\wp(B)=\big\{a,b,\varnothing,\{a\},\{b\},B\big\}\;.$$
If at some point you study more advanced set theory, you’ll find that in $\mathsf{ZF}$, the set theory that is the de facto standard for axiomatic set theory, all objects are sets. Each of the numbers $\pi,\sqrt2,47$, and $666$ of my example is a set, though unless one is doing something highly set-theoretic or foundational one needn’t worry about the actual structure of those sets. In everyday mathematics one doesn’t (and needn’t) think of them in those terms, so in everyday mathematics we do tend to think in terms of types of objects: numbers, sets of numbers, ordered pairs of numbers, etc. However, it’s still possible to lump together different types (in this sense) of mathematical objects into sets like $\Bbb R\cup\{\varnothing\}$, but we rarely have much reason to do so — again, outside of purely set-theoretic contexts.
