Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
3  
Note that we do not have $xy \geq 1$ if one or both of $x, y$ is zero, and zero is an integer. –  Old John Jan 6 '13 at 16:58
    
If x or y is zero, that xy≥1 will be false, so (x,y)∉R. –  Freewind Jan 6 '13 at 17:01
    
Yes, exactly that. –  Old John Jan 6 '13 at 17:02
2  
@Freewind: You forgot about $0$, easy to do, it is so small. The relation is not reflexive because $(0,0)\not\in R$. For all $x\ne 0$, $(x,x)\in R$. So reflexivity almost holds. –  André Nicolas Jan 6 '13 at 17:04
2  
I understand now! I misunderstood the definition of reflexive: A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.. There is every element a ∈ R, so we must consider 0. –  Freewind Jan 6 '13 at 17:04

2 Answers 2

up vote 5 down vote accepted

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.

share|improve this 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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.