What Determines the Transitivity of Relations in a Set?

  • Thread starter Thread starter nuuskur
  • Start date Start date
  • Tags Tags
    Relations Set
Click For Summary
SUMMARY

The discussion centers on the concept of transitivity in relations within a set, specifically addressing the definition: If (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R. It clarifies that for a relation to be transitive, all combinations of elements in the set must be tested for violations of this definition. The empty set is highlighted as inherently transitive since there are no elements to violate the transitive property, making any relation on it transitive.

PREREQUISITES
  • Understanding of set theory and relations
  • Familiarity with logical implications in mathematics
  • Knowledge of the definition of transitivity
  • Basic skills in testing mathematical properties
NEXT STEPS
  • Explore the properties of equivalence relations in set theory
  • Learn about the implications of reflexivity and symmetry in relations
  • Study examples of transitive and non-transitive relations
  • Investigate the role of the empty set in mathematical proofs
USEFUL FOR

Students of mathematics, educators teaching set theory, and anyone interested in the foundational concepts of relations and their properties.

nuuskur
Science Advisor
Messages
926
Reaction score
1,220
Having trouble understanding the concept of transitivity.
By definition: If (a,b)\in R\wedge (b,c)\in R \Rightarrow (a,c)\in R - Great.

Consider the set \{a,b\}. What makes the relation \{(a,a)\} or \{(a,a),(a,b)\} transitive? How do I translate this in terms of the definition?
What makes an empty set transitive?
 
Physics news on Phys.org
There is an important part missing in the definition: "For all a,b,c in the set".
You can test all combinations of the set and see if this statement is violated for one combination. If yes, the relation is not transitive. If there is no violation, it is transitive.
For the empty set, there is no combination at all that could violate transitivity, so a relation on it is always transitive.
 
  • Like
Likes nuuskur

Similar threads

MHB Give a set and a relation that satisfies the properties
  • · Replies 10 ·
Replies
10
Views
2K
MHB Satisfying S∘S=S: What Are The Conditions?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Supremum of a set, relations and order
  • · Replies 35 ·
2
Replies
35
Views
5K
MHB Proving Reflexive, Symmetric and Transitive Properties of Relation R on P(U)
  • · Replies 3 ·
Replies
3
Views
2K
MHB Equivalence Relations: Solving & Proving Reflexivity, Symmetry & Transitivity
  • · Replies 2 ·
Replies
2
Views
2K
MHB Upper Bound of Sets and Sequences: Analyzing Logic
  • · Replies 3 ·
Replies
3
Views
5K
MHB Can you explain how the law of logic was used to reach this conclusion?
  • · Replies 3 ·
Replies
3
Views
3K
MHB Problem understanding relations syntax.
  • · Replies 1 ·
Replies
1
Views
1K
MHB Understanding Binary Relations: Reflexive, Symmetric, Antisymmetric & Transitive
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad If symmetric then transitive relation
  • · Replies 4 ·
Replies
4
Views
2K