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:
“With two sons born eighteen months apart, I operated mainly on automatic pilot through the ceaseless activity of their early childhood. I remember opening the refrigerator late one night and finding a roll of aluminum foil next to a pair of small red tennies. Certain that I was responsible for the refrigerated shoes, I quickly closed the door and ran upstairs to make sure I had put the babies in their cribs instead of the linen closet.”
—Mary Kay Blakely (20th century)