Buod Ng Tigre Tigre

12 December 2016

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, the  function contains the list , indicating that integer successor of  is . A relation is another kind of interrelationship among objects in the universe of discourse. More formally, a relation 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 list , indicating that  is less than . 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.

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

For example, the list  is a member of the  function, and there is no other list of length 2 with  as its first argument, i. e. there is only one successor for . By contrast, the < relation contains the lists , , and so forth, indicating that is less than , , 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 lists  and , reflecting the fact that the sum of  and  is  and the fact that the sum of  and  and  is . Similarly, the relation < contains the lists  and , reflecting the fact that  is less than  and the fact that  is less than  and  is less than . This flexibility is not essential, but it is extremely convenient and poses no significant theoretical problems. Relation:  A relation is simply a set of ordered pairs. | A relation can be any set of ordered pairs.

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