Q:

An example of a semiring which is not a ring.

In Bogachev's book on measure theory he states the following:

The family of all intervals in the interval $[a,b]$ gives an example of a semiring that is not a ring.

I am sure this is an easy task to do; I just want to check if I am correct or not.
If we look at $[a,c]$ $a<c<b$ then $[a,c] \setminus [a,c) = \{ c\}$, now I assume that sets as $\{ c \}$ aren't regarded in this context as intervals (even though we can write them as some interval with a weak inequality); so we get that this difference isn't in this space so it's not a ring; the fact that it's a semiring I am not sure how to show this, maybe since if $[x,y] \setminus [x0 ,y0]$ where $[x0,y0] \subset [x,y] \subset [a,b]$, then this equals to:
$[x,x0) \cup (y0,y]$ as required, the other sorts of intervals are the same just need to alter between $[ ]$ or $( )$.

A:

Let $a< c<d< b$.
Then $(a,b)\setminus (c,d)=(a,c]\cup [d,b)$, which is clearly not an interval.
Thus the set of intervals is not closed under set difference.

