# Can a reflexive relation be a strict superset of the identity relation?

I read on wikipedia that R is a reflexive relation if xRx. It also said that a reflexive relation is a non-strict superset of the identity relation, so my question is, can a reflexive relation be a strict superset of the identity relation? I'm wondering this because I also read about coreflexive relations and they can be a strict subset of the identity relation, example: X is a set of natural numbers and R means "x,y from X are equal odd numbers", here R is a strict subset of the identity relation.

• The identity relation is $I = \{ (x,x) \mid| x \in A\}$ for some set $A$. A reflexive relation $R$ on $A$ contains at least these ordered pairs, so $R$ is a superset of $I$. $R$ will be a strict superset iff it contains at least one other ordered pair. Sep 17, 2015 at 5:55
• Additional: $A$ is a superset of $B$ if $B$ is a subset of $A$. Sep 17, 2015 at 5:56
• Sure: if $A$ is the underlying set for the relation, $A\times A$ is reflexive, and if $A$ has more than one element, it’s a strict superset of the identity relation on $A$. Sep 17, 2015 at 5:56
• Actually, I am confused again, at first I thought about putting 2 different elements into the relation, but then realized that we have to only put one. Is there an example? And @BrianM.Scott I don't understand what you meaning by AxA is reflexive, if it's cartesian product, then if A = {1,2} then AxA={(1,1),(1,2),(2,1),(2,2)} or do you mean the subset {(1,1),(2,2)} of AxA? Sep 17, 2015 at 8:06
• @paulpaul1076: A relation on $A$ is reflexive if it contains every ordered pair $\langle a,a\rangle$ for $a\in A$. In your example, $A\times A$ does indeed contain $\langle 1,1\rangle$ and $\langle 2,2\rangle$, so $A\times A$ is reflexive. So are the relations $\{\langle 1,1\rangle,\langle 1,2\rangle,\langle 2,2\rangle\}$ and $\{\langle 1,1\rangle,\langle 2,1\rangle,\langle 2,2\rangle\}$: both of them contain both of the identical pairs $\langle 1,1\rangle$ and $\langle 2,2\rangle$. And of course the identity relation on $A$, which contains only those two pairs, is also reflexive. Sep 17, 2015 at 8:09