Define a Relation on S X S* by...

I have spent the past several hours trying to wrap my mind around this question. It should be noted that this is homework and that I am not asking for anyone to do the work for me. But I need some help to figure out what I should be doing.

I will begin by stating the question as it is written:

"In this exercise we will be working mod 15. Notice that in mod 15 there are *zero-divisors*, that is, pairs of non-zero elements a and b with ab = 0; for example, 3 x 5 = 10 x 6 = 0, etc. Let S = {0, 1, ..., 14} and let S* denote the set of non-zero elements of S.

Define a relation ~ on S X S* by: (a, b) ~ (c, d) <===> there exists a zero divisor *u* with u(ad - bc) = 0."

I began this problem by creating the set S* and filling it with the non-zero elements of S. I did this by taking two elements of S (a, b) and determined whether ab mod 15 = 0. Thus I created the set S* of ordered pairs: S* = { (3, 5), (3, 10), (6,5), (9,5), (6,10), (5, 12), (10,12) } (This may or may not be correct).

so with this framework in mind I have attempted to answer the following:

a) Show that (2, 10) ~ (1, 5) This question threw me off right away since (1, 5) isn't in my S* set. But plugging the values into the equation I got;

u(ad - bc) = 0

u(10 - 10) = 0 for all u

b) Show that (2, 10) ~ (2, 13) Plugging the values into the equation I go;

u(ad - bc) = 0

u(26 - 20) = 0

u(6) = 0

u = 10 or u = 5.

For these first two problems I have no idea if I'm doing this properly this is just what I have gotten based on my understanding of the question. If I am incorrect in anything please help me to understand.

c) show that ~ is reflexive and symmetric: For this question I have no idea even how to begin. The first thing I notice is that in order for it to be reflexive, for every element a in S (a, a) is in ~. But this doesn't hold because if a = 1 then (1, 1) doesn't appear in the S* set because 1x1 = 1 mod 15 = 1 and thus is not a zero-divisor. Does this then prove that this relation is NOT reflexive?

and finally d) find all pairs (a, b) such that (a, b) ~ (1, 5). Once again on this one I'm not sure where to start.

I would appreciate any help in wrapping my brain around this stuff. I want to understand.

Re: Define a Relation on S X S* by...

S* is just S-{0}, i.e. {1,2,3,.. 14}

you sort of stumbled your way through (a) and (b)

for (c) show that $(a,b)\sim (a,b)$ and that

$(a,b)\sim (c,d),~(c,d)\sim (e,f) \Rightarrow (a,b)\sim (e,f)$