-1
$\begingroup$

This post is an extension of my previous post.

Suppose I have an Excel (or a csv or whatever 'paper printed table') with 2 columns: $A$ and $B$.

Example:

 
-------
A  | B
-------
1   2.5
7   5.5
8   9.7
1   2.7

For each row I would like to forbid the following: $a\neq b$.

So for instance in row 1 I cannot have $a=3$ and $b=3$. However, the question is not how to implement it in Excel. The question is how to use the correct math notation to express this constraint.

Let's suppose the domain of $A$ is $\mathbb{N}$ and the domain of $B$ is $\mathbb{R}$. (that's the difference between this post and the previous post).

On my previous post user 'valdim123' proposded :

$$\forall n\in N, An\neq Bn$$

However in the previous post both $A$ and $B$ had the same domain $\mathbb{N}$.

So how to make sure we compare the same rows if $A$ and $B$ have different domain? If I mimimic 'valdim123' notation the following proposal is, in my opinion, incorrect because $n\neq r$ (for instance I don't want to compare row 2 of col A and row 3 of col B) I need to set $n=r$ but I do not get how to write that...

$$\forall n\in N,\forall r\in R, An\neq Br$$

  • 0
    This doesn't make any sense. The Excel entry A3 means the column is A and the row is 3. A is not a variable, it is a constant, drawn from the universe of possible columns. – 2017-01-10
  • 0
    Indeed, so how to write formally this constraint? (I mean when domain of $A$ is $\mathbb{N}$ and domain of $B$ is $\mathbb{R}$) – 2017-01-10
  • 1
    Okay, my answer to your previous question addresses what you want. $N$ is not the natural numbers, it is the set of rows. In your first example, $N=\{1,2,3,4\}$. $A1=1, A2=7, B1=2.5, B2=5.5$. You need to abandon $a,b$, they are confusing you. The entries are named $XY$, where $X$ denotes the column and $Y$ denotes the row. – 2017-01-10
  • 0
    Ah OK thanks a lot for clarification now I understand what $N$ means. – 2017-01-10
  • 0
    @vadim123 I think transfer your comment as the answer. So I can mark it as done. – 2017-01-11

1 Answers 1

1

As mentioned in the comments, the issue is that $a$ is not a well-defined variable. Each entry is described by a column (either A or B), and a row (chosen from a set of rows). It is the row number that needs to be quantified over.