Transitivity as property of relation

In summary: Similarly, a relation can be symmetric, but this term only makes sense when applied to a specific relation. Therefore, "transitivity" is not standalone by itself.
  • #1
roni1
20
0
In Hebrew, one explain to me that:
"Transitivity is a property (or attribute - I don't which word is correct) of property".
So,
(1) Which word is correct?
(2) Why Transitivity is not standalone by itself?
(3) Are there relations of other kind, that no standalone by themselves?
 
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  • #2
roni said:
(1) Which word is correct?
Either "property" or "attribute" can be used. These words are used in their usual sense from the dictionary, not in a technical sense. The second occurrence of "property", though, is probably used in a technical sense.

From the mathematical standpoint, a property, or predicate, on a set $A$ is any subset of $A$. Alternatively, a predicate on $A$ can be defined as a function from $A$ to the set {true, false}. The the subsets consists of those elements of $A$ that are mapped to true.

A binary relation on a set $A$ is any subset of $A\times A$, a Cartesian product of $A$ and $A$, which is a set of all ordered pairs of elements of $A$. Thus, a binary relation is a property of ordered pairs, or a predicate on $A\times A$. Binary relations can be transitive or not. Thus, "transitivity" is a property of a binary relation, i.e., something that can be true or not about this relation.

roni said:
(2) Why Transitivity is not standalone by itself?
Any adjective, like "transitive", requires a noun in mathematics just like in everyday language.

roni said:
(3) Are there relations of other kind, that no standalone by themselves?
An adjective "continuous" is a property of functions in calculus, so to talk about continuity you must apply it to some functions.
 

FAQ: Transitivity as property of relation

1. What is transitivity as a property of relation?

Transitivity is a property that describes the relationship between three elements in a set. It states that 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.

2. Why is transitivity important in scientific research?

Transitivity is important in scientific research because it allows researchers to make inferences and draw conclusions based on indirect relationships between variables. This can help to uncover hidden patterns and connections that may not be immediately evident.

3. How is transitivity used in mathematical models?

In mathematical models, transitivity is often used to represent complex relationships between multiple variables. By incorporating transitivity, the model can account for indirect effects and make more accurate predictions.

4. Can transitivity be violated?

Yes, transitivity can be violated in certain cases. This can occur when there are conflicting relationships between elements in a set, or when there are exceptions to the general trend. In these cases, the indirect relationship between elements may not hold true.

5. How does transitivity relate to other properties of relations?

Transitivity is closely related to other properties of relations, such as reflexivity and symmetry. Reflexivity states that an element is always related to itself, while symmetry states that if element A is related to element B, then element B is also related to element A. Transitivity can be seen as an extension of these properties, as it takes into account the indirect relationships between elements.

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