Understanding Relations in Mathematics

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Discussion Overview

The discussion revolves around understanding mathematical relations, specifically focusing on properties such as reflexivity, symmetry, antisymmetry, and transitivity. Participants explore these concepts through examples involving circles and recurrence relations, seeking to clarify definitions and their applications.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a relation on circles defined by the center of one circle being inside another and questions its reflexivity, symmetry, antisymmetry, and transitivity.
  • Another participant reiterates the same examples and prompts for clarification on what has been done so far and the understanding of definitions.
  • Some participants express confusion regarding the definitions provided in their textbooks compared to the examples discussed, indicating a struggle with the phrasing of the problems.
  • One participant suggests rephrasing the properties in terms of the specific sets being discussed, particularly focusing on symmetry and reflexivity.
  • Another participant emphasizes the need for clarity on the definitions of relations and encourages sharing learned definitions to facilitate understanding.

Areas of Agreement / Disagreement

Participants do not appear to reach consensus on the definitions and applications of the properties of relations. There is a mix of confusion and differing interpretations of the concepts presented.

Contextual Notes

Participants express uncertainty regarding the definitions of relations and how they apply to the examples given. There is mention of differing educational backgrounds and potential misunderstandings of the terminology used.

liahow
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Not really understanding these concepts.

Consider the following relation on the set of all circles in the xy plane: A ~ B if and only if the center of circle A is inside circle B. Is ~ reflexive? Is ~ symmetric? Is ~ antisymmetric? Is ~ transitive? Prove answers.


Consider the following relation on the set of all recurrence relations. X ~ D if and only if all of the terms of the sequence associated with D appear in the sequence associated with X. Is ~ reflexive, symmetric, antisymmetric, transitive? Prove.
 
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liahow said:
Not really understanding these concepts.

Consider the following relation on the set of all circles in the xy plane: A ~ B if and only if the center of circle A is inside circle B. Is ~ reflexive? Is ~ symmetric? Is ~ antisymmetric? Is ~ transitive? Prove answers.


Consider the following relation on the set of all recurrence relations. X ~ D if and only if all of the terms of the sequence associated with D appear in the sequence associated with X. Is ~ reflexive, symmetric, antisymmetric, transitive? Prove.
What have you done so far ? Do you understand the definitions ?
 
As far as the definitions go, what we learned in our books is different from this equation. We've been given sets and told to go with those. Perhaps it's just my own ignorance, but when a problem is so dramatically phrased differently than the way I learn it, I am stuck in neutral.
 
liahow said:
...
Consider the following relation on the set of all circles in the xy plane: A ~ B if and only if the center of circle A is inside circle B. ... Is ~ symmetric?
...
Let's look at this one. Rephrase the abstract property "symmetric" in terms of the set you are working with. In this case, it is asking "If a circle A has its center inside circle B, does it necessarily follow that circle B has its center inside circle A ?"
Do this with the rest of the questions. :smile:
 
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hypermorphism said:
Let's look at this one. Rephrase the abstract property "reflexive" in terms of the set you are working with. In this case, it is asking "If a circle A has its center inside circle B, does it necessarily follow that circle B has its center inside circle A ?"
Do this with the rest of the questions. :smile:
Just so liahow isn't confused, the question above is for the symmetric property. A ~ B if and only if the center of circle A is inside circle B, so reflexive would just be A ~ A, or "the center of circle A is inside circle A".
 
liahow said:
As far as the definitions go, what we learned in our books is different from this equation. We've been given sets and told to go with those. Perhaps it's just my own ignorance, but when a problem is so dramatically phrased differently than the way I learn it, I am stuck in neutral.
You were asked "what is the definition" of a relation. Please tell us what definitions you have learned, whether they are "different from this equation" or not (I don't quite understand that- you haven't cited any equations).
 

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