foundations of computational agents
The mathematical concepts we build on include:
A set has elements (members). We write if is an element of set . The elements in a set define the set, so that two sets are equal if they have the same elements.
An -tuple is an ordered grouping of elements, written . A 2-tuple is a pair, and a 3-tuple is a triple. Two -tuples are equal if they have the same members in the corresponding positions. If is a set, is the set of -tuples where is a member of . is the set of -tuples where each is in .
A relation is a set of n-tuples. The tuples in the relation are said to be true of the relation. An alternative definition is in terms of the relation’s characteristic function, a function on tuples that is true for a tuple when the tuple is in the relation and false when it is not.
A function, or mapping, from set , the domain of , into set , the range of , written , is a subset of such that for every there is a unique such that . We write if .
While these may seem like obscure definitions for common-sense concepts, you can now use the common-sense concepts comfortable in the knowledge that if you are unsure about something, you can check the definitions.