Q:

Cardinal of the quotient set

Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. In $X \times X$ define the equivalence
$(a,b)\:\mathcal{R}\:(c,d)$ if $a+b=c+d$. Then what is the cardinal of the quotient set?
I know that $|X| = 9$, so $| X \times X | = |X| |X| = 81$. But how use the relation for the cardinal of the quotient set? 
Thank you.

A:

Hint
Let $r = a+b$ with $a,b\in X$
How many different values can $r$ have?

