a) Consider the relation R defined on Z(the integers) by x~y when xy is greater than or equal to 0. Is R an equivalence relation?
b) Would your answer to (a) change if R was defined on Z by x~y when xy is greater than 0?
I often think about equivalence relations in terms of the following fact. Let X, Y be sets and f a function from X to Y. Define a binary relation R on X as follows: R(x, y) iff f(x) = f(y). Then R is an equivalence relation. This can be straightforwardly proved by definition.
Considering (a), can we find a function f such that x ~ y iff f(x) = f(y)? The fact that x ~ y, i.e., xy ≥ 0, approximately means that x and y have the same sign, i.e., x and y are both positive or both negative. I said "approximately" because there are problems when x or y is zero. Can a positive number be related to 0? What about a negative number? What does this say about transitivity?
Concerning (b), x ~ y does mean that x and y have the same sign. Let
Then x ~ y iff sgn(x) = sgn(y). Unfortunately, we can't conclude that ~ is an equivalence relation using the fact above because sgn is not a function from Z: it is not defined on 0. Which property of an equivalence relation does this break?