Symmetry, reflexivity and transitivity in set relations I am really having a difficult time applying the definitions of the above three set relations terms. 
For the following problem $R = \{(x,y)|xy \geq 1, x, y \in Z\}$ I have to determine whether the expression is reflexive, symmetric, antisymmetric and/or transitive.
According to the book's answer, it is not reflexive because, for $(a, a) \in Z$ where $a = 0$, $0 \times 0 \ngeq 1$.
However, it states that the expression is symmetric, i.e. $(a,b) \in R \to (b,a) \in R$ for $a \neq b$. But I don't understand how this could hold, because if either $a$ or $b$ are zero then clearly the expression would be false.
I also don't understand how to apply the rule of transitivity to this expression; though the definition states $(a,b) \in R \land (b,c) \in R \to (a,c) \in R$, I don't have a variable $z$ in the expression to work with. 
I know I am not understanding this or doing this correctly, but the book pretty much just gives definitions without many examples, as if it should be obvious (well it is not, for thick-headed folk like me).

Again, for transitivity the key word is if: 
if $\,(a,b)\,,\,(b,c)\in R\,\,\,,\,\,\text{then also}\,\,(a,c)\in R\,$ .
So, if $\,ab\geq 1\,\,\,\text{and also}\,\,\,bc\geq 1\,,\,\text{then}\,\,ac\geq 1$ ,
otherwise we'd have
$$ac<1\Longrightarrow ac=0\,\,\,or\,\,\,ac\leq -1$$
as all the elements are integers, but then either
$$(1)\,\,a=0\,\,\,or\,\,\,c=0\Longrightarrow (a,b)\notin R\,\,\,or\,\,\,(b,c)\notin R\,\,,\,contradiction\,\,,\,\text{or}$$
$$(2)\,\,a>0\,,\,c<0\,\,\,or\,\,\,a<0\,,\,c>0$$
I'll leave it to you to show that any of the last two cases in (2) leads to contradiction, too.
