Relation $R$ is $xy\ge1$, and $x\in \mathbb{Z}$ and $y\in\mathbb{Z}$, is $R$ reflexive?

Question

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 if

b) $xy \ge 1.$

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.

Answer 1

Recall that $xRy$ if and only if $(x,y)\in R$, and $R$ is defined to be reflexive if and only if $(x,x) \in R$ for all $x$.

In this case, $0$ is an integer, and the statement $xy \geq 1$ is false if one or both of $x, y$ is zero, so your relation fails to be reflexive.

Answer 2

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.