What is the Definition of a Relation in Mathematics?

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The definition of a relation in mathematics lacks consensus, with some sources defining it as a set of ordered pairs and others as a subset of a Cartesian product. The discussion highlights the nuances between these definitions, emphasizing that they are not equivalent. One definition assumes the ability to construct sets from ordered pairs, while the other operates within an existing set framework. A proposed definition clarifies that a relation consists of ordered pairs if and only if the elements belong to the relation. The conversation reflects a desire for clearer foundational agreements in mathematical definitions.
quantum123
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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?
 
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The two definitions say the same thing.
 
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.
 
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.
 
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Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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