What is the Definition of a Relation in Mathematics?

[quantum123]

It is kinda strange. There is no agreement on the definition of a relation.
Some books says it is a set of ordered pairs.
Other books says it is a subset of a cartesian product.
How nice if everything can be agreed down to a few axioms like Euclid's elements.

What is your favourite definition of a relation?

 

[micromass]

The two definitions say the same thing.

 

[quantum123]

I do not agree with that. In the definition using ordered pairs, it is assumed that a set can be built from ordered pairs. But in the cartesian definition, a set is provided already, you just use a part of it via subset.

 

[xxxx0xxxx]

I go with:

$$R \mbox{ is a relation} \Leftrightarrow \forall r(r \in R \Rightarrow \exists x \exists y (r=<x,y>))$$

This definition states that the elements of R are ordered pairs if and only if R is a relation.

 

[blue_raver22]

I understand the importance of having clear and agreed upon definitions in order to accurately communicate and understand concepts. In the case of relations, it is true that there is no universal agreement on its definition. However, this does not necessarily mean that it is a strange concept. In fact, this lack of agreement can be attributed to the complexity and versatility of relations.

My favorite definition of a relation is the one that defines it as a set of ordered pairs. This definition is simple and intuitive, and it allows for the understanding of relations as a way to connect or relate elements from two sets. Additionally, it aligns with the concept of a function, which is a specific type of relation that is widely used in mathematics and science.

While it would be nice to have a single agreed upon definition for relations, the diversity of definitions also reflects the multifaceted nature of this concept. As scientists, it is important for us to understand and be open to different perspectives and definitions, as they can offer valuable insights and approaches to studying and understanding relations.