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

  • Thread starter brookey86
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  • #1
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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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Answers and Replies

  • #2
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So am I on the right track?
 
  • #3
D H
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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.
 
  • #4
HallsofIvy
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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?
 
  • #5
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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.
 
  • #6
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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.
 
  • #7
D H
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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.
 
  • #8
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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.
 

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