If symmetric then transitive relation

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Discussion Overview

The discussion revolves around the properties of relations, specifically examining the implications of symmetry and transitivity. Participants explore whether the condition of symmetry (if xRy and yRx) implies transitivity (if xRy and yRz then xRz) and discuss examples to illustrate their points.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that if xRy and yRx, then xRx could imply transitivity, using the example of the set {(x,x),(y,y)} as both symmetric and transitive.
  • Another participant provides a counterexample with the relation R = {(x,x),(y,y),(z,z),(x,y),(y,x),(y,z),(z,y)}, which is symmetric and reflexive but not transitive, highlighting that xRy and yRz does not guarantee xRz.
  • Some participants argue that the implication of xRy and yRx leading to xRx is a weaker statement than what is required for transitivity, suggesting that transitivity may necessitate at least three distinct elements in a set.
  • It is noted that even if a relation is transitive, there may be no instances of x and y such that both xRy and yRx hold, as illustrated by the order relation '>'.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between symmetry and transitivity, with no consensus reached on whether symmetry implies transitivity. The discussion remains unresolved regarding the implications of these properties.

Contextual Notes

Participants acknowledge that the implications of symmetry and transitivity may depend on the specific elements of the set being considered, and that certain examples may not fulfill the conditions necessary for transitivity.

rajeshmarndi
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Isn't, if we have xRy and yRx then xRx will also make transitive? Because if I am right {(x,x),(y,y)} on set {x,y} is symmetric and transitive.

Isn't the above similar to, if xRy and yRz then xRz is transitive relation?

Thanks.
 
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##R = \{(x,x),(y,y),(z,z),(x,y),(y,x),(y,z),(z,y)\}##
is an example of a relation that's symmetric and reflexive without being transitive, because
##xRy \leftrightarrow yRx##
and
##yRz \leftrightarrow zRy##
but we also have ##xRy \wedge yRz## without ##xRz##.
 
Similar but not nearly as strong. xRy and yRx => xRx is a statement about a much smaller set of x and y than the transitive property requires.
 
FactChecker said:
Similar but not nearly as strong. xRy and yRx => xRx is a statement about a much smaller set of x and y than the transitive property requires.
Does this mean transitive relation require atleast 3 distinct element of a set e.g {x,y,z}.

Also as I mentioned, is {(x,x),(y,y)} on set {x,y} reflexive along with symmetric and transitive.
 
(xRy and yRx) => xRx only makes a statement about the x & y where both xRy and yRx. There might easily be none of those, so it might say nothing.

PS. Even if the relation R is transitive, there may be no x & y where (xRy and yRx). An example is the order relation '>'. It's not possible for (x > y & y > x), even though '>' is a transitive relation.
 

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