Let \(S\) denote the set of all ordered pairs of real numbers; that is, \(S = \{(a,b) : a,b\in \mathbb{R}\}\).
Given \((a_1,b_1), (a_2,b_2) \in S\), \((a_1,b_1)+(a_2,b_2)\) is defined as follows: \[(a_1,b_1)+(a_2,b_2)=(a_1-a_2, b_1 b_2).\] Here, \(a_1-a_2\) is the usual difference between \(a_1\) and \(a_2\) and \(b_1b_2\) is the usual product of \(b_1\) and \(b_2\).
Is the operation \(+\) as defined above associative? In other words, is it true that for all \(u, v, w\in S\), \(u+(v+w)= (u+v)+w\)?
Yes
No