A function is one kind of interrelationship among objects. For every finite sequence of objects (called the arguments), a function associates a unique object (called the value). 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. 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. it 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.