In 'Mathematics Form and Function' ch.2, section 4, 'Integers' by Saunders Mac Lane (p.50 in the 96 edition) I came across the following definitions of sum and product for ordered pairs:
(m, n) + (m', n') = (m + m', n + n')
(m, n) (m', n') = (mm' + nn', mn' + m')
My understanding is that each m or n are related to m' and n' in such a way that, for example, if m' is m + 1, then n' must be n + 1; the relationship must be consistent between both the original and subsequent numbers.
Given that, if I plug in some real numbers such as m = 2 and n = 3 (and in this case the number with the prime symbol is equivalent to adding 1), I would get this:
(2 + 3, 3 + 4) = (5, 7) # still obtain an ordered pair, so far so good
((2 * 3) + (3 * 4), (2 * 4) + 3) = (18, 11) # not an ordered pair!
My understanding is the result of the product should be an ordered pair with the 2nd element greater than the 1st, but that's the not case because 18 > 11. Is this an error, typo, or am I fundamentally misunderstanding the maths here?
Thanks for any advice you can give.
The notion 'ordered pair' simply refers to a sequence of any elements of length 2, independently from any ordering among the elements.
So, $(18,\,11)$ is a perfect ordered pair.
'Ordered' here only means that the order of the two elements is important, so that $(18,\,11)\ne(11,\,18)$.
Also, in the definition, $m, n, m', n'$ are all arbitrary, there is no assumption on any relations among them.