What Determines the Transitivity of Relations in a Set?

In summary, transitivity is a property of a relation where if (a,b) and (b,c) are in the relation, then (a,c) must also be in the relation. This can be tested by considering all combinations of elements in the set and checking for any violations of this property. An empty set is always transitive because there are no elements to create a violation.
  • #1
nuuskur
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Having trouble understanding the concept of transitivity.
By definition: If [itex](a,b)\in R\wedge (b,c)\in R \Rightarrow (a,c)\in R[/itex] - Great.

Consider the set [itex]\{a,b\}[/itex]. What makes the relation [itex]\{(a,a)\}[/itex] or [itex]\{(a,a),(a,b)\}[/itex] transitive? How do I translate this in terms of the definition?
What makes an empty set transitive?
 
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  • #2
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.
 
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FAQ: What Determines the Transitivity of Relations in a Set?

What is a transitive relation in a set?

A transitive relation in a set is a mathematical concept that describes the relationship between three elements in a set. If element A is related to element B, and element B is related to element C, then element A is also related to element C. In other words, if A is connected to B and B is connected to C, then A is indirectly connected to C.

How is a transitive relation represented?

A transitive relation is usually represented as a set of ordered pairs, where each pair has the form (A, B) and indicates that element A is related to element B. All of the ordered pairs in the set must follow the transitive property, meaning that if (A, B) and (B, C) are both in the set, then (A, C) must also be in the set.

What is an example of a transitive relation?

An example of a transitive relation is the "greater than" relation in numbers. If A is greater than B and B is greater than C, then A is also greater than C. For instance, if 5 > 3 and 3 > 1, then 5 > 1.

Can a transitive relation be reflexive or symmetric?

No, a transitive relation cannot be reflexive or symmetric. A reflexive relation is one where every element is related to itself, while a symmetric relation is one where if A is related to B, then B is also related to A. Since transitive relations require a chain of connections between elements, they cannot be reflexive or symmetric.

Why are transitive relations important?

Transitive relations are important in mathematics and other fields because they allow us to make logical inferences and draw conclusions based on indirect connections. They also play a crucial role in defining other mathematical concepts, such as equivalence relations and partial orders.

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