Next, we check

. This means checking to make sure that for all integers x, y, we have x ? y ? y ? x. Unwind the definition of ? as we have done for (1) and we see that x ? y ?

= 4 m for some integer m y ? x ?

= 4 m for some integer m Based on that, is (2) true? If so, enter Y; if not, enter a pair of integers for which this is false.

Finally, we check (3). This means checking to make sure that for all integers x, y, z, if x ? y and y ? z then x ? z. Is this true? If so, enter Y; if not, give a triple of integers for which this fails.

Finally, based on this calculation, is ? an equivalence relation on the set of integers? Enter Y or N.