I am currently taking real analysis, and I'm afraid this might be a very elementary question, but the further I go in the course the more confused I become about the difference between a set in $\mathbb{R}^p$ and a subset of $\mathbb{R}^p$.
Furthermore, sometimes the notes simply say, for example, "$U_n \subseteq \mathbb{R}^p $". Is $U_n$ a subset or a set?
Finally, what is the notation for a set and a subset, respectively?
Thank you in advance!
Best Answer
By definition, a set is simply a collection of elements.
By definition, a set $A$ is a subset of a set $B$ if and only if every element in $A$ is also in $B$. In other words, $A \subseteq B$ iff $x \in A$ implies $x \in B$.
Note that every set is in fact a subset. Because, trivially, $A \subseteq A$ for all sets $A$. In words, "every set is a subset of itself."
Also, the empty set $\varnothing$ is a subset of every set $A$, because the empty set vacuously satisfies the definition of subset. In words, "the empty set is a subset of every set."
Oftentimes questions will ask you for examples of nontrivial subsets. This typically means any subset that is neither the entire set itself nor the empty set.
"A set in $\Bbb R^p$" and "a subset of $\Bbb R^p$" mean exactly the same thing.
To further expand on that:
If $A$ is "a set in $\Bbb R^p$" then that means all the elements of $A$ are also in $\Bbb R^p$. So $A$ satisfies the definition of being a subset of $\Bbb R^p$, and we have $A \subseteq \Bbb R^p$.
If $A$ is "a subset of $\Bbb R^p$" then, well, this is just a more direct way of saying exactly the same thing: $A \subseteq \Bbb R^p$
Both. $U_n$ is itself a set. And $U_n$ is a subset of $\Bbb R^p$. That's actually exactly what the notation means. When you see something like "$A \subseteq B$" you can read it as "$A$ is a subset of $B$."
I'm not entirely sure what you're getting at here but there really isn't any notation for a set, at least not in the same way that there is for subsets. We generally denote sets with capital letters (except for the empty set, typically denoted $\varnothing$ or $\emptyset$), but this isn't really a hard rule.
Subset-related notation:
$A \subseteq B$ means $A$ is a subset of $B$.
$A \supseteq B$ means $A$ is a superset of $B$ (which is just another way of saying $B$ is a subset of $A$).
Some authors use $\subset$ and $\supset$ instead of $\subseteq$ and $\supseteq$, respectively.
The notations $A \nsubseteq B$ and $A \not\subset B$ can be used to indicate $A$ is not a subset of $B$. There are corresponding superset notations $\nsupseteq$ and $\not\supset$.
The notation $A \subsetneq B$ means $A$ is a proper subset of $B$. This just means that $A \subseteq B$ and $A \ne B$. There is a corresponding superset notation $\supsetneq$.
Note: Unfortunately, some authors (although I think this is rare these days) use $A \subseteq B$ to indicate $A$ is a subset of $B$ and use $A \subset B$ to indicate $A$ is a proper subset of $B$.