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I am studying preference relations in fuzzy logic with regards to quantifying consistency of preferences. In particular, how to quantify the consistency of somebodies preferences between three options, $i, j, k$.

My preference between any two variables can be any value in the domain $0\leq r \leq 1$

If I like $i$ more than $j$, then $r_{ij}$ is > 0.5

If I like $j$ less than $k$, then $r_{jk}$ is < 0.5

If I can't choose between $i$ and $j$ then $r_{ij} = 0.5$

Also, $r_{ij} = 1-r_{ji}$

If $r_{ij} = 0.7$ (arbitrary example) and $r_{jk} = 0.4$ then there is some function $f$ which determines my preference $r_{ik}$, if I can be modelled as perfectly rational.

$$r_{ik} = f(r_{ij}, r_{jk})\forall i,j,k$$ Where $f$ is a binary operator of the form: $$f:[0,1]\times[0,1]\to[0,1]$$

I keep seeing these functions described as "unique up to a positive linear transformation".

My issue is that I don't really have any idea what this means. I am a mechanical engineer, have done A-level maths, and a little university level maths as part of my engineering degree. Is it possible to explain roughly what this means, In a way that I would be able to understand. Appreciate any help anyone can offer.

Alex Ortiz
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James Izzard
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1 Answers1

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This question already has an answer in other forms on this site, under the guise of "What does the phrase 'up to [...]' mean in mathematics?" To answer you directly, in general when we say that an object $f$ in mathematics is unique up to $x$ where $x$ depends on the context, we mean that $f$ is uniquely determined, except possibly for the addition of some $x$.

Examples help:

  1. For instance, if $f'$ is the derivative of some function $f$ on $[a,b]$, then we can say that $f$ is unique up to a constant because if $f' = g'$, then $f = g + C$ for some constant $C$.

  2. If we are classifying groups of order $4$, then we would say that there are exactly two groups of order $4$, up to isomorphism. So, we can make a list $G,H$ of two nonisomorphic groups of order $4$, such that if any other group $N$ has order $4$, then there is an isomorphism $\varphi$ such that $\varphi:N\cong G$ or $\varphi:N\cong H$, but not both.

  3. In your example, for a function $f$ to be "unique up to a positive linear transformation," it likely means that $f$ is only determined up to the addition of some positive linear map $T$, so talking about $f$ itself is sort of meaningless because we can only talk about $f$ "up to $T$". Just like saying "the antiderivative of $f'$ is $f$" is meaningless, because we can only talk about the antiderivative of $f'$ up to a constant $C$.

Alex Ortiz
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