Calculating the quotient set numbers on the relation $(a,b)R(c,d)\iff a+d=b+c$

Question

Let $A = \{0,...,10\}$ and $R$ be a binary relation on $A\times A$ defined by

$(a, b) R (c, d)$ if and only if $a + d = b + c$

How many equivalence classes are there in the quotient set $A^2/R$?

Any assistance will be appreciated. Thanks.

Answer 1

Hint:

Rewrite that relation as $$(a,b)R(c,d)\iff a-b=c-d$$or equivalently as:$$(a,b)R(c,d)\iff f(a,b)=f(c,d)$$where $f:A^2\to\mathbb Z$ denotes the function prescribed by $(a,b)\mapsto a-b$

This reveals that the number of equivalence classes is the same as the cardinality of the image of $f$.