Pair and Tuple
A binary operation, ab, depends on the ordered pair (a, b) and so (ab)c (where the parentheses here mean first operate on the ordered pair (a, b) and then operate on the result of that using the ordered pair ((ab), c)) depends in general on the ordered pair ((a, b), c). Thus, for the general, non-associative case, binary operations can be represented with binary trees.
However:
- If the operation is associative, (ab)c = a(bc), then the value depends only on the tuple (a, b, c).
- If the operation is commutative, ab = ba, then the value depends only on { {a, b}, c}, where braces indicate multisets.
- If the operation is both associative and commutative then the value depends only on the multiset {a, b, c}.
- If the operation is both associative and commutative and idempotent, aa = a, then the value depends only on the set {a, b, c}.
Read more about this topic: Binary Operation
Famous quotes containing the word pair:
“Not the less does nature continue to fill the heart of youth with suggestions of his enthusiasm, and there are now men,—if indeed I can speak in the plural number,—more exactly, I will say, I have just been conversing with one man, to whom no weight of adverse experience will make it for a moment appear impossible, that thousands of human beings might exercise towards each other the grandest and simplest of sentiments, as well as a knot of friends, or a pair of lovers.”
—Ralph Waldo Emerson (1803–1882)