I have a problem to do that is similar to this:
$R_1$ is over the set of real numbers
(a) $(x, y) \in R_1$ if and only if $xy = 5$
decide whether it is reflexive, anti-reflexive, symmetric,
anti-symmetric and transitive.
I'm confused, I know that reflexive means x=x and symmetric means that x,y implies y,x. I think it's the format of the question that is throwing me off. Help is very much appreciated.
Best Answer
Is $R_1$ reflexive? That means: Does $xx=5$ hold for every real number $x$?
Is $R_1$ anti-reflexive? Does $xx\ne5$ hold for every real number $x$?
Is $R_1$ symmetric? Does $xy=5$ imply $yx=5$ for real numbers $x,y$?
Is $R_1$ transitive? Does $xy=5$ & $yz=5$ imply $xz=5$ for real numbers $x,y,z$?
First, $R_1$ is not reflexive, because $17$ is a real number and $17\cdot17\ne5$.
Next, $R_1$ is not anti-reflexive, because $\sqrt5$ is a real number and $\sqrt5\cdot\sqrt5=5$.