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Write a short informal English description of the following set

$\{w|w$ is a string of 0s and 1s and w equals the reverse of $w \}$ Ans: The set of all strings of 0s and 1s

Write formal descriptions of the following sets:

The set containing the empty string

Ans: {a,b,c} because any set must contain the empty string, but I think it's actually {e} because I am thinking of empty string as empty set. I am not sure

The set containing nothing at all.

Ans: That one is a little tricky since every set at least contains the empty set, nevertheless I am not sure if the empty is known as an element of any set

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In your informal description, you don't mention the important condition that the string must be the same backwards and forwards.

It's not true that any set must contain the empty string. Your second suggestion $\{e\}$ is right provided $e$ is your notation for the empty string.

The set containing nothing at all sounds like an informal description of the empty set $\emptyset.$ It's not true that every set contains the empty set.

Perhaps this confusion is cause one could interpret "$S$ contains $X$" as $X\in S$ (i.e. $X$ is an element of $S$) or $X\subseteq S$ ($X$ is a subset of $S$). It seems here like they mean the former. It's true that for any set $S$ that $\emptyset \subseteq S,$ but is not necessarily (and usually not) true that $\emptyset\in S.$

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1) Your informal description includes all strings. The set you were trying to describe doesn't include the string 01, for example, because 01 is not the same backwards as forwards. So your informal description can't be describing the right set. You need to include the reversability condition in your description.

2) The empty string is just another string - it doesn't have any special status when it comes to set membership. The empty set, for example, does not contain the empty string - just like an empty bucket doesn't have an empty box in it. If $e$ denotes the empty string, then $\{e\}$ is the set you're looking for.

3) Every set has the empty set as a subset, but that's not what "contains" means here. Here, "contains" means "has as a member". Not every set contains the empty set; for example, the set $\{1, 2\}$ doesn't have the empty set in it (its only members are $1$ and $2$). The set containing nothing at all is just the set with nothing in it - one way of writing that would be $\{\}$.

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    How do you interpret the following set $\{n|n$ is an integer and $n=n+1\}$. At first I thought it's the set of all integers, but now I think about it the expression $n=n+1$ is a condition, but this condition is impossible to fulfill, so it's the empty set right? – 2017-01-21
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    @TheMathNoob Correct, that's the empty set. – 2017-01-21