# Thread: is this relation reflexive?

1. ## is this relation reflexive?

The relationship R on Z is defined by $\displaystyle (a,a) \in R$ if and only if a-b is odd. Is this relation reflexive?

I said no because a-a=0 and 0 is even.

I'm having trouble getting my head wrapped around reflexive, is this right?

2. ## Re: is this relation reflexive?

Originally Posted by Jskid
The relationship R on Z is defined by $\displaystyle (a,b) \in R$ if and only if a-b is odd. Is this relation reflexive?
I said no because a-a=0 and 0 is even.
You are correct.
Reflexive simply means that each element is related to itself.

3. ## Re: is this relation reflexive?

the "gold standard" for reflexive relations is: aRb iff a=b (the diagonal relation). a relation S is reflexive iff R (the diagonal relation) is a subset of S, that is if S includes the diagonal.

so the relation ≤ is reflexive, but < is not (neither is >). for your relation, we can draw a partial picture like so:

4 * _ * _
3 _ * _ *
2 * _ * _
1 _ * _ *
...1 2 3 4 ...the *'s indicate pairs in the relation, and the _'s indicate pairs not in the relation. as you can see (visually) nothing on the diagonal is in the relation.