For each of the following relations defined on the positive integers:
$>, <, =, ≥, ≤$
How do I justify the relation is reflexive, symmetric, or transitive?
I only know that $=$ is reflexive because $<1,1><2,2><3,3><4,4><5,5>$ where $N=N$
but they are nether $>$ nor $<$. But It could be $≥, ≤$ because of the equal sign.
Now for symmetric, I understand that it would be $<1, 1><1,2><2,2><2,1><1,3><3,3><3,1><1,4><4,4><4,1><1,5><5,5><5,1>$ Where it has the same element as reflexive that $N=N$. But i'm not sure all relations here are symmetric. It seems they all have met the requirements, since $<1, 2>$ where $1$ is less than $2$, and $<2, 1>$ where $2$ is greater than $1$.
For transitive, I'm just confused. I understand that if step 1 can go to step 2, and step 2 can go to step 3, then step 1 can go to step 3. But how does it work here? $<1,1><1,2><2,2><2,3><1,3><3,3><3,4><2,4><4,4><4,5><3,5>$
Again $<1, 2>$ where $1$ is less than $2$, and $<2, 1>$ where $2$ is greater than $1$.
All I understand now is the relation:
$=$ is all reflexive, symmetric, and transitive.
$≥, ≤$ is reflexive
$>, <$ (symmetric and transitive?)
\begin{array}{|c|c|c|c|c|c|}
\hline & Reflexive & Symmetric & Transitive\\ \hline
= & Yes & Yes & Yes \\ \hline
< & No & (Maybe?) & (Maybe?) \\ \hline
> & No & (Maybe?) & (Maybe?) \\ \hline
≤ & Yes & (Maybe?) & (Maybe?)\\ \hline
≥ & Yes & (Maybe?) & (Maybe?)\\ \hline
\end{array}
Best Answer
A relatio $R$ is reflexive if for all $x$ we have $xRx$, as you've pointed out. This is the case for $=, \geq, \leq$, but not for $<, >$, as you've also pointed out.
You've misunderstood what symmetric means. A relatio $R$ is symmetric if $xRy$ impplies $yRx$. Said a bit differently, a relation is symmetric if it is completely and utterly irrelevant which number is to the right and which number is to the left of the relation symbol. This is the case for $=$, but none of the four others.
A relation is transitive if $xRy$ and $yRz$ implies $xRz$. All your relations statisfy this. Examples: $2<5$ and $5<9$, therefore $2<9$. Or this ome: $$ 2=\text{The probable number of arms on a healthy human being}\\ \text{The probable number of arms on a healthy human being} = \text{The number of objects in a pair} $$ implies $2 = \text{The number of objects in a pair}$. But these things are so obvious it's difficult to grasp what transitiveness means.
More instructive, then perhaps, is an example of a relation which is not transitive. One non-transitive relation is $R = \text{"differs by at most five from"}$. We have $1R3$ and $3R8$, but it is not the case that $1R8$.