# How to write down formally number(s) of occurence?

I am trying to see how to have more than one condition using Iverson bracket notion. Basically, I am trying to represent this if statement formally.

Count number of occurence when [ (A<x) and (B<y)] is true.

I am aware of the following question.

My first guess was:

$$C(x) = \sum_{i=1}^n [A < x][B < y]$$

which I am not sure if true.

x and y are arbitrary numbers, while A and B values are attributes of each element in a set where the size of the set is n. The pseudocode for the above statement is:

for i in n:
if i.A < x and i.B < y:
count++


where count is the total number of time the condition was true.

• What do you mean by "number of occurrences" in this context. Is there a set of points $(x,y)$ under consideration? Is there an indexed list of pairs $(x_i,y_i)$? Are $x$ and $y$ fixed but $A$ and $B$ independently can take any of a set of values? etc. The answers to these will determine how best to write things down. Nov 25, 2021 at 16:10
• Hi @MarkS. thanks for reply. I added more details. Nov 25, 2021 at 16:52

First of all, while you use i.A and i.B in your code, there is no connection between $$A$$, $$B$$ and $$i$$ in your formula. Now, in math, “objects” that have properties you can access are not a normal way of setting things up, so you wouldn’t normally see $$i{.}A$$ or $$i_A$$ or $$i(A)$$. Instead, it’s more common to arrange the A-values and B-values into sequences, i.e. to write $$A_i$$ and $$B_i$$ (or $$A(i)$$ and $$B(i)$$, if you prefer). You can index your A-values and B-values by integers, but other indexing sets might work, too.
Your loop then becomes $$\sum_{i = 1}^n [A_i < x] [B_i < y].$$ I included the transformation from $$[\text{A_i < x and B_i < y}]$$ to $$[A_i < x] [B_i < y]$$ that you did in your post. Either version is fine.
Last, it is a bit strange that you used $$C(x)$$ even though the value depends on both $$x$$ and $$y$$. (I would have expected $$C(x,y)$$.) But this might be fine, if $$y$$ is generally fixed whereas $$x$$ varies.