# XRy: x has drawn a picture of y | what relations apply?

## Homework Statement

The relation xRy is defined as "x has drawn a picture of y". R is on the set of all people.

Is this relation: reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive ?

## Homework Equations

What confuses me about this problem is that there is uncertainty involved. If xRy had been defined as "x scored higher on a test than y", then we could definitively say that y did not score higher than x, x did not score higher than himself, etc. However in this case, R doesn't definitively conclude that y did or did not draw a picture of x, or if x also drew a picture of himself, etc.

## The Attempt at a Solution

reflexive - no; can't conclude that x drew x
irreflexive - no; can't conclude that it's never the case that x drew x
symmetric - no; can't conclude that y also drew x
asymmetric - yes; in some case both xRy and yRx could be true
antisymmetric - no; if x drew y then we can't always say that y did not draw x
transitive - no; we don't know if yRw means x drew w

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So am I on the right track?

D H
Staff Emeritus
To say that an operator has such a characteristic (e.g., reflexive, irreflexive, ...), the characteristic need to hold for all x, y (or all x, y, z if applicable), not just a select handful. In other words, one counterexample is all it takes to answer have the answer be "no". For example, that someone could paint a self portrait means the relationship cannot be irreflexive. That not everyone has painted a self portrait means the relationship cannot be reflexive.

HallsofIvy
Homework Helper
In some texts, "asymmetric" simply means "not symmetric" in others, it means "if aRy then we do NOT have yRx". Which does your text use?

In some texts, "asymmetric" simply means "not symmetric" in others, it means "if aRy then we do NOT have yRx". Which does your text use?
Our text uses the latter.

To say that an operator has such a characteristic (e.g., reflexive, irreflexive, ...), the characteristic need to hold for all x, y (or all x, y, z if applicable), not just a select handful. In other words, one counterexample is all it takes to answer have the answer be "no". For example, that someone could paint a self portrait means the relationship cannot be irreflexive. That not everyone has painted a self portrait means the relationship cannot be reflexive.
So it seems I'm correct with my reasoning above, except perhaps for asymmetric because x drawing y doesn't necessarily imply that y did not draw x.

D H
Staff Emeritus