# Multiset indicator function

I'm writing a report and was wondering wheter my notation is understandable? I'm fairly new to using the maths notation. A similarity measure between a multiset $u$ and a set $c$ is defined as:

$sim_2(u,c) = \frac{1}{|c^t| + |u^t|} {\sum_{i=1}^N 1_{\bf A} : u_i^t \to \mathbb{Z}^\geq}$

where $N$ is the number of items in $u^t$ and $1_{\bf A} : u_i^t \to \mathbb{Z}^\geq$ is the multiset indicator function.

The part that I'm not sure about is the multiset indicator function. Just came across it and am not sure if I am using it correctly? The idea being to count number of times the $i^{th}$ element appears in $u^t$ if and only if that element appears in $A$.

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1A:uti→Z≥ looks like a function definition to me but i'm not sure written as i've written it it is entirely correct. if u trust my limited mathematical knowledge u should use this to define the indicator function but then u don't need the whole function definition in your equation. i think you can say something like this.

$sim_2(u,c)=...\sum 1_{\bf A}$

where $1_{\bf A} : u_i^t \to {0,1}$

in words this says "the similarity measure between u and c equals the number of elements in $u^t$ (cardinality) that satisfy the function $1_{\bf A}$, where the function $1_{\bf A}$ takes an element of ut and returns a 0 or 1". i understand that's what your trying to say?

anyway, the important thing to make clear in your definition is that the indicator function maps a single element of ut (i.e. $u_i^t$) to {0,1}. i.e. given some element uti the indicator function returns either a 0 or 1 dependent on whether the element is in A. it is when this is summed over all N that we get a positive integer (i.e. $\sum 1_{\bf A} \to \mathbb{Z}^\geq$).

you could alternatively wrap the whole definition up like this: $sim_2 : u X c \to \mathbb{Z}^\geq$. this says, the function $sim_2$ takes 2 inputs (u and c) and returns a positive integer. and then include the full function definition: $sim_2(u,c)=\frac{1}{∣∣ct∣∣+∣∣ut∣∣}\sum 1_{\bf A}$

i think what i'm saying is correct. hope it helps. you might want a second opinion though. not from a doctor just a real mathematician.

good luck!

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thanks -- one question though. according to wikipedia in the multiset indicator function the constraint that $1_A : u_i^t --> {0,1}$ is relaxed so that the values become the positive integers - in my head that then makes what I've written correct. en.wikipedia.org/wiki/Multiset –  Matti Lyra Apr 28 '11 at 17:02