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

In summary, the relation xRy is defined as "x has drawn a picture of y". R is on the set of all people. Reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive?
  • #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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  • #2
So am I on the right track?
 
  • #3
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
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
HallsofIvy said:
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
D H said:
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
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
D H said:
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.
 

1. What is XRy in the context of drawing a picture?

XRy is a notation used in mathematical set theory to represent the relationship between two elements, where X is the subject and Y is the object. In this case, it signifies that X (the person) has drawn a picture of Y (the object).

2. How is this relationship different from other types of relationships?

XRy specifically denotes the act of drawing a picture, which is a creative and artistic process. Other types of relationships, such as X is taller than Y or X is the parent of Y, are more objective and do not involve creativity.

3. What does "relations apply" mean in this context?

"Relations apply" refers to the connections or links that can be made between X and Y in the drawing. These could include visual relationships, such as X is looking at Y, or symbolic relationships, like X is representing Y in the picture.

4. How is this concept relevant in scientific research?

In scientific research, XRy can be used to analyze and understand relationships between different elements. For example, it could be used to study the relationship between an artist and their subject, or to examine the connections between different objects in a painting.

5. Can XRy be applied to other forms of art or media?

Yes, XRy can be used in any creative medium where one element is representing or depicting another. This could include mediums such as music, film, sculpture, and more.

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