comparing two tuple or set or list with greater than (>) or less than operator(<)

Question

I have two set/list/tuple. A=(A1,A2,A3) and B=(B1,B2,B3). I know that each element of A is greater than B, meaning A1>B1, A2>B2 and A3>B3. How can I write this in correct mathematical expression or notation/representation. Can I write A>B?

Answer 1

Tuples (or lists) are a particullar kind of set, so it's usefull to distinguish between a set as general and a tuple or list.

Set

You use a set if you don't care about the order and you don't care if there are repetitions, also, you use brackets, an example is

$$\{a,b,c\} = \{a,b,a,c,c\}$$

List and tuples

You use a list or tuple if you do care about the order and repetition and you usually use parentheses, in this case

$$(1,2,3) \neq (1,1,3,2,2) $$

Both are usefull to denote what you want

Let $A = (a_1,a_2,a_3)$ and $B = (b_1,b_2,b_3)$ be tuples of real numbers (formally you would say $A,B \in \mathbb{R}^3$). As far as I know there is not a (universally used) symbol "$>_*$" such that "$A >_* B$" means that $a_1 > b_1$, $a_2 > b_2$, $a_3 > b_3$. Although, you can express that idea saying the following:

Let $A = (a_1,a_2,a_3)$ and $B = (b_1,b_2,b_3)$ be tuples of numbers such that $a_i > b_i$ for any $i \in \{1,2,3\}.$

This last example method is very useful since it can be generalized to any tuple of $n$ elements:

Let $A = (a_1,a_2,...,a_n)$ and $B = (b_1,b_2,...,b_n)$ such that $a_i > b_i$ for any $i \in \{1,2,...,n\}.$

This way you can define without ambiguity and in general the previous symbol "$>_*$".

Definition. Let $A = (a_1,...,a_n) \in \mathbb{R}^n$ and $B = (b_1,...,b_n) \in \mathbb{R}^n$, we define

$$A >_* B \text{ if and only if } a_i > b_i \text{ for any } i \in \{1,2,...,n\}.$$

Answer 2

The partial order $\preceq$ on ${\bf R}^n$ defined by $$(x_1, \ldots, x_n) \preceq (y_1, \ldots, y_n) \mbox{ if and only if } x_1\leq y_1, \ldots, \mbox{and } x_n \leq y_n$$ Is often called the dominance relation on ${\bf R}^n$. But it is just one of any number of orders one could define. So you certainly can use >, but if you don’t explain your usage (i.e., which partial order you mean) it will be ambiguous.