# Buod Ng Tigre Tigre

More formally, a function is defined as a set of finite lists of objects, one for each combination of possible arguments. In each list, the initial elements are the arguments, and the final element is the value. For example, theÂ Â function contains the listÂ , indicating that integer successor ofÂ Â isÂ . A relation is another kind of interrelationship among objects in the universe of discourse. More formally, aÂ relationÂ is an arbitrary set of finite lists of objects (of possibly varying lengths). Each list is a selection of objects that jointly satisfy the relation.

For example, the < relation on numbers contains the listÂ , indicating thatÂ Â is less thanÂ . Note that both functions and relations are defined as sets of lists. In fact, every function is a relation. However, not every relation is a function. In a function, there cannot be two lists that disagree on only the last element. This would be tantamount to the function having two values for one combination of arguments.

By contrast, in a relation, there can be any number of lists that agree on all but the last element.

For example, the listÂ Â is a member of theÂ Â function, and there is no other list of length 2 withÂ Â as its first argument, i. e. there is only one successor forÂ . By contrast, the < relation contains the listsÂ ,Â , and so forth, indicating thatÂ is less thanÂ ,Â , and so forth. Many mathematicians require that functions and relations have fixed arity, i. e they require that all of the lists comprising a function or relation have the same length. The definitions here allow for functions and relations with variable arity, i. e. t is perfectly acceptable for a function or a relation to contain lists of different lengths. For example, the + function contains the listsÂ Â andÂ , reflecting the fact that the sum ofÂ Â andÂ Â isÂ Â and the fact that the sum ofÂ Â andÂ Â andÂ Â isÂ . Similarly, the relation < contains the listsÂ Â andÂ , reflecting the fact thatÂ Â is less thanÂ Â and the fact thatÂ Â is less thanÂ Â andÂ Â is less thanÂ . This flexibility is not essential, but it is extremely convenient and poses no significant theoretical problems. Relation:Â Â A relation is simply a set of ordered pairs. | A relation can be any set of ordered pairs.