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:
“To get a man soundly saved it is not enough to put on him a pair of new breeches, to give him regular work, or even to give him a University education. These things are all outside a man, and if the inside remains unchanged you have wasted your labour. You must in some way or other graft upon the man’s nature a new nature, which has in it the element of the Divine.”
—William Booth (1829–1912)