Ordered tuples and Cartesian products
The Cartesian product of the sets S and T, denoted by S×T, is the set
of all ordered pairs (s,t) where s?S and t?T. In other words,
S×T = {(s,t) | s?S ? t?T}
If in sets the order of elements does not matter, in ordered n-tuples
(a1,a2,…,a) it does. So does the number of repetitions of an element:
(a1,a2,…,an) = (b1,b2,…,bm) iff n=m and a1=b1, a2= b2,…, an= bm.
So that (a,b)?(b,a); (a,b,b)?(a,b)
The Cartesian product of the sets S1, S2, …, Sn, denoted by
S1 × S2 × … × Sn, is the set of all ordered n-tuples (s1,s2,…,sn) where
s1? S1, s2? S2, …, sn? Sn. In other words,
S1×S2×…×Sn = {(s1,s2,…,sn) | s1? S1 ? s2? S2 ? … ? sn? Sn}