Let A= {x is reals:x>0} and define a relation on A by x relation y If xy=0 for x,y in A .
I was wondering if this is reflexive relation. So far I thought If x=1 and y= 0, then 1*0=0 and 0*1 is also =0. It can be reflexive not sure if I am doing it right by using this counter example . How do I prove what sort of relation it is?
On
Let us call R the relation you are considering is.
By definition,
R = { (x,y) belonging to R² | x . y = 0}
Let us think of this relation as a machine that selects pairs of real numbers in the plane R². In other words, a machine that selects points in the plane R².
Certainly this machine selects all points that have 0 as X-coordinate. In other words, all points belonging to the line x=0 are selected. ( This line it nothing else but the Y-axis).
The machine also selects all points that have 0 as Y-coordinate. In other words , all points belonging to the line y=0 are selected. ( This line is the X-axis itself).
In order this relation to be reflexive, it should select all points on the line whose equation is y = x.
Is it the case?
No, the only point of the line with equation y=x that is selected by our " R-machine" is the point (0,0) .
Pick at random any other point of the line with equation y=x, say the point
( 0, 00001; 0, 00001).
Is this point " selected" by our relation.
It would be selected IFF 0, 00001 times 0, 00001 is equal to 0 ( Remember this is the sufficient and necessary condition to be " selected" by our "R-machine").
But obviously, this condition is not fullfilled! 0,00001 times 0,00001, as you know, is NOT equal to 0.
Conclusion: failing to select all the points belonging to the line y=x, our relation is NOT reflexive.
It is not reflexive. If $x>0$, then $x\times x>0$. In particular, $x\times x\neq0$.