(Solved): ( 3 points) In this problem we work out step-by-step the procedure for checking an equivalence rel ...

( 3 points) In this problem we work out step-by-step the procedure for checking an equivalence relation. Denote by $$\mathbb{Z}$$ the set of all integers. Declare that two integers $$x,y$$ are related if $$x+y$$ is an integer multiple of 8 . In symbols: $$x?y?8\text{ divides }x+y.$$ We want to check if this is an equivalence relation. That means we need to check if $$?$$ is (1) Reflexive (2) Symmetric (3) Transitive We begin with (1). This means checking to make sure that for all integers $$x$$, we have $$x?x$$. Recall the definition of $$?$$ for this problem and we see that this is equivalent to saying that for all integers $$x$$, we have that $$(x+x)$$ is an integer multiple of 8 . Is this true? If so, enter Y; if not, enter an integer for which this is false. Next, we check (2). 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 $$x?y?=8m$$ for some integer $$m$$ $$y?x?=8m$$ 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 $$\mathrm{Y}$$ or $$\mathrm{N}$$.