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:
“There are two kinds of truth; the truth that lights the way and the truth that warms the heart. The first of these is science, and the second is art.... Without art science would be as useless as a pair of high forceps in the hands of a plumber. Without science art would become a crude mess of folklore and emotional quackery.”
—Raymond Chandler (1888–1959)