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.

Why is the relation R on A irreflexive if and only if ΔA ∩ R = ∅?

I always thought the empty set is reflexive (and transitive, symmetric because it is vacuously true.)

share|improve this question
4  
The empty relation is reflexive and also irreflexive. –  Peter Franek Jun 22 at 21:35
    
Thank you for your comment. I did not know that. After reading Hagen von Eitzen's comment as well and searched further, I started to understand. Thanks again. –  user3125591 Jun 22 at 21:56

1 Answer 1

up vote 4 down vote accepted

By definition, $R$ is irreflexive iff for all $a\in A$ we have $(a,a)\notin R$. This precisely says that the diagonal $\Delta A=\{\,(a,a)\mid a\in A\,\}$ is disjoint from $R$. Note that according to this, the empty relation is irreflexive, and it ias also (vacuously) symmetric and transitive. But the empty relation is reflexive only as relation on the empty set (i.e. $R=\emptyset$ is reflexive iff $A=\emptyset$).

share|improve this answer
    
I understand it now! Very clear explanation! Thank you very much. You explained it more clear than what's in my book! –  user3125591 Jun 22 at 21:54

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.