Difference between $⊂$ and $⊆$?

I was solving the exercises in Discrete Mathematics and its applications book.

Determine whether each of these statements is true or false.

1. {0} ⊂ {0}
2. {∅} ⊆ {∅}

I thought both 1 and 2 are true, but when I checked the answers I found that 1 is false and 2 is true.
I got confused and distracted because I don't know the difference between them.

Solution 1:

If the book distinguishes between $\subset$ and $\subseteq$, then most likely the former symbol denotes proper inclusion, so $\{0\}\subset\{0\}$ is false. The latter symbol instead will denote inclusion (with possible equality).

However it's very common to find $\subset$ denoting inclusion (with possible equality), so one always has to check or try and infer from the context. Don't take Wikipedia pages as revealed truth.

It's so common that $\subset$ denotes nonstrict inclusion that somebody uses $\subsetneq$ to denote proper inclusion, for safety.

Solution 2:

There are two conventions and one hybrid:

You can either use

I)

i) $A \subset B$ means $A$ is a subset of $B$ and $A$ might equal $B$.

ii) $A\subsetneq C$ means $A$ is a subset of $B$ but $A \ne B$.

II)

i) $A \subseteq B$ means $A$ is a subset of $B$ and $A$ might equal $B$.

ii) $A\subset C$ means $A$ is a subset of $B$ but $A \ne B$.

Your book uses II). I personally prefer $I$ (and judging by your post you do too, it would seem). The other posters and wikipedia claim II) (which your book uses) is more common and accepted. That may be true but I'm not going to make that claim.

Anyway. If you use convention I both 1 and 2 are true. If you use convention II 1 is false and 2 is true.

(Because $\{0\}$ is a subset but is equal to $\{0\}$ and $\{\emptyset\}$ is a subset but is equal to $\{\emptyset\}$.

III) Just to avoid confusion we can do this hybrid solution that has not ambiguity:

i) $A \subseteq B$ means $A$ is a subset of $B$ and $A$ might equal $B$.

ii) $A\subsetneq C$ means $A$ is a subset of $B$ but $A \ne B$.

Just to be safe. No-one can mistake what we mean. I'd recommend using III) just to avoid this silly quibbles.

(It's actually not that bad. It's usually very clear in context which ones make sense.)

==== addendum ====

A comment points out that if the author uses two different symbols that implies the author wants the symbols to mean two different things. And by convention I) there is no symbol $\subseteq$ so it's reasonable that the author is using connvention II).

Which makes me think further about my opinion of which convention I prefer.

And I think a little concern should be made for the ease of reading and writing.

"$\subset$" means "is a subset of" and an immediate impression is on seeing it is not not really think or worry about whether it is a proper subset or not. So I believe if you want to indicate the possibility of inequality must be considered and thought about or that the possibility of equality is not an option, one should indicated this further by specifically indicating so with $\subseteq$ or $\subsetneq$. If a reader sees simply $\subset$ it is, in my opinion, not reasonable to expect the reader to consciously consider or know that equality is not possible.

So I'd go with hybrid. For reading, and trying to second guess an author I'd view in context.