Pairwise comparing two sequences (notation)

Lets assume we have to sequences with equal number of variables $A = \{a_1, a_2,..., a_n\}$ and $B = \{b_1, b_2,..., b_n\}$. I need to compare each value pairwise: $a_1$ to $b_1$, $\cdots$, $a_n$ to $b_2$; and decide if at least one of the values of $B$ is larger than its corresponding value in $A$. It the a way to writte it short mathematically?

Solution 1:

The elements of two sets cannot be compared as you described even if they have the same number of elements because there is no specific pairing law. So the comparing does not make sense.

I assume $A$ and $B$ are $n$-tuples (sequence) denoted by $A = (a_1, \cdots, a_n)$ for $A$.

Then, the formal expression would be $\newcommand{\N}{\mathbb{N}}$ $$ \exists N \in \N: (N \le n \land b_N > a_N) $$ or simply $$ \exists N \le n: b_N > a_N $$ read there is $N$ such that $b_N > a_N$. You can check whether the sentence is true or not from $A$ and $B$.

Comment: We use $\{\}$ to denote a set and $()$ to denote a sequence or $n$-tuple.