This is a question from book "Discrete Mathematics and Its Applications".
9.1.7
Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where(x, y)
∈ R if and only ifb) $xy \ge 1.$
The answer provided by the book is: $R$ is symmetric and transitive.
Why isn't $R$ reflexive?
I think $R$ is reflexive because $x$ and $y$ are integers, since $xy \ge 1$, they are positive integers or negative integers, that $xx \ge 1$ should be true, that $R$ should be reflexive.
Best Answer
A relation $\,R\,$ on a set $A$ is reflexive if and only if, for every element $\,a\in A,\,$ it is true that $\;(a, a) \in R\;$.
In this problem, suppose $x = 0$. Since $\;0 \in \mathbb{Z}$, but $xx = 0\cdot 0 \ngeq 1$, thus $(0, 0) \notin R$, and so the relation fails to be reflexive on the set of integers.
It only takes one element as a counterexample to prove the relation on the set non-reflexive.
If your relation were defined on the set of non-zero integers, then it would be reflexive.