Understanding Reflexive Relations

Question

I'm reviewing some problems to try and get a better understanding of relations. I get how a reflexive relation works on a defined relation with numbers, but not so much when its done with a set builder of sorts.

Let $R$ = {$(a,b)|a + b$ is odd} be a relation on all integers.

My professor's answer to $R$ being reflexive is that "Let $a$ =2, 2 + 2 = 4 and 4 is not odd. Therefore, ℛ is not reflexive."

For a function to be reflexive if for all $x\in A$, $(a,a)\in R$, this is where my flaw of understand this comes in but doesn't that mean that unless the relation is defined then all relations are reflexive?

EDIT: for a relation being defined I meant this $R$ on {1,2,3} given by $R$ = {(1,1), (2,2), (2,3), (3,3)}.

Answer 1

No. The condition for reflexivity for a relation $R$ is that for all $a$ in the domain of $R$, $(a, a) \in R$. If there is a single element $a$ in the domain such that $(a, a) \not\in R$, $R$ is not reflexive.

Answer 2

Consider R = {(a,b)| a < b} be the relation "is less than". This is a relation. Any two numbers are either related this way or they are not. This relation is not reflexive because 1 is not less than 1.

Actually, your professor just gave you a good counter-example (2,2) $\notin$ R whether R ={(a,b)| a + b = odd} or R = {(a,b)| a < b}.

Now consider R = {(a,b)| a + b = even} then R is reflexive because for all a, a+a = 2a is even.

Or consider R = {(a,b)| a = m*b where m is integer}. Then 4 R 2 but 2 notR 4. But for any a; a = 1*a so a R a, so R is reflexive (but not symmetric).

And here's an example where 3 R 3 but 2 not R 2. R = {(a,b)|a and b odd}. (So not reflexive.)

And finally yours were R is reflexive and "nothing else" R = {(a,b)| a = b} = {(a,a)}

Answer 3

Here is some serious hand-waving aiming at giving some intuition.

See the cartesian product $A \times A$ as some sort of "generalized grid", i.e. There is one row for all $a \in A$ as well as and one column for all $a \in A$.

If $(a,b) \in R$ then blacken the entry at row $a$ and column $b$.

$R$ is reflexive if and only if all entries on the main diagonal are blackened. Bingo !