What is the difference between non-reflexive or irreflexive?

Question

What is the difference between a non-reflexive and irreflexive relation? Is one stronger than the other (i.e. all non-reflexive relations are irreflexive or vice-versa)?

Answer 1

Is this a trick question? [Original question: What's the difference between "reflexible" and "irreflexible"?] None: in both cases, there's no such thing :) You mean "reflexive" and "irreflexive".

A relation $R \subseteq A \times A $ is reflexive on $A$ if $aRa$ for every $a\in A$. Thus $R$ is not reflexive on $A$ iff for some $a\in A, \text{not } aRa$.

A stronger condition than "not reflexive" is irreflexive. $R$ is irreflexive on $A$ iff for all $a\in A, \text{not } aRa$.

If $A\neq \emptyset$, then "$R \text{ is irreflexive on } A$" implies "$R \text{ is not reflexive on } A$", but the converse is not in general true unless $A$ is a singleton.

Answer 2

On set $A=${$1,2,3$}, consider the relations

$(1)$ $R_1=${$(1,1),(2,2),(3,3),(1,2)$}

$(2)$ $R_2=${$(1,1),(2,2),(1,2)$}

$(3)$ $R_3=${$(1,2),(2,1),(3,1)$}

$R_1$ is reflexive,$R_3$ is irreflexive and $R_2$ is non-reflexive. Can you see the difference?