# Set notation, what is the meaning of <dimension>[]?

I just ran across this notation in a lecture slide and unfortunately it was not explained,

$R$ is a relation of membership values in a 2D universe, and $u$ is the membership value.

1. What is the hat over $y$ meant to indicate?

I know the hat over $R$ means that it is transitive, and the dot over $R$ means that it's max-product transitive, but I am not sure what a hat over a dimension means.

2. What do the square brackets to the right of $y$ hat mean?

3. What do the underlines under $R$ mean?

My guess is this means "$\forall y, u_R(x,z) \Leftarrow$ (the left hand side)", but I want to be sure.

I get the feeling that this equation is missing something but unfortunately this is all I've been given and I am having trouble finding more information on Min-Sub Distance Resemblance Relations.

Also, I would appreciate links to your sources (books/web pages/etc.), I would like to get more detailed information on set theory.

After much digging I found the answer in an old book

The equation I was given was incorrectly copied to the lecture slides.

The y with a hat over it is actually supposed to be a ⋀ (min operator) with a y under it, meaning "min over all y" for the entries inside the square brackets.

The Rs with the lines under them are incorrectly typed, they should be Rs with a squiggly line underneath just meaning that R contains fuzzy data.

My source:

"Introduction to the Theory of Fuzzy Subsets", Volume 1, by A. Kaufman, (C) 1975.

This book can't be found on amazon, it's rather rare.