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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?

CS1
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2 Answers2

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It is not reflexive. If $x>0$, then $x\times x>0$. In particular, $x\times x\neq0$.

José Carlos Santos
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  • sorry I am little confused on how prove what sort of relation it is . – CS1 Mar 17 '18 at 11:19
  • Wasn't your question simply asking whether it is reflexive or not? Did I not answer that question? – José Carlos Santos Mar 17 '18 at 11:28
  • Yes u did answer it thanks. – CS1 Mar 17 '18 at 11:31
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    @CS1 That is not the most adequate way to thank for a good answer. You should consider to mark it as correct and/or upvote it. Those are the ways of acknowledging the work others have done for you. Moreover, whenever you accept an answer, you also gain 2 points of reputation. That' often how it starts. See here – amrsa Mar 17 '18 at 11:48
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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.