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

[brookey86]

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

 

[brookey86]

So am I on the right track?

 

[D H]

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]

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?

 

[brookey86]

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.

 

[brookey86]

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]

Correct. Now that you have clarified what "asymmetric" means, they are all false. That both xRy and yRx can be true (i.e., x and and y can draw pictures of each other) means that it is not asymmetric.

 

[brookey86]

Correct. Now that you have clarified what "asymmetric" means, they are all false. That both xRy and yRx can be true (i.e., x and and y can draw pictures of each other) means that it is not asymmetric.

Thank you for the help. I couldn't find any online texts that explained relations where uncertainty was involved.