30.3 Input and Output Patterns

Input and Output Patterns are the recognizable arrangements by which input values and output values can correspond to one another within a relation, cataloguing every basic way that repetition or uniqueness among inputs and outputs can appear, and showing precisely which of these arrangements the Function Criterion permits and which it forbids.

Distinct Inputs with Distinct Outputs describes the pattern in which every input value in the relation is unique, appearing only once across the entire Ordered Pair Collection, and every output value is likewise unique, so that no value in either position repeats anywhere within the relation. This pattern automatically satisfies One Output for Each Input, since with no input ever repeating, there is no opportunity for Repeated Input Inspection to uncover any conflict at all.

Distinct Inputs with a Shared Output describes the pattern in which every input value remains unique, but two or more of those distinct inputs happen to be paired with the identical output value. Because each input still appears only once, this pattern likewise satisfies One Output for Each Input without difficulty, since the criterion places no restriction whatsoever on how many different inputs may share a common output.

Many Inputs to One Output is the broader description of the pattern illustrated by Distinct Inputs with a Shared Output, generalized to any number of distinct input values converging on the same single output value, remaining fully compatible with Function Classification regardless of how many inputs participate in this convergence.

One Input to Multiple Outputs describes the pattern in which a single input value appears paired with two or more genuinely different output values across separate ordered pairs in the collection, directly matching Same Input with Different Outputs and constituting the one pattern explicitly forbidden by the function criterion, since it violates One Output for Each Input outright.

Repeated Output Permitted restates, as a general principle drawn from Distinct Inputs with a Shared Output and Many Inputs to One Output, that repetition among output values carries no consequence for whether a relation qualifies as a function, regardless of how many times a particular output value recurs across different, distinct input values within the collection.

Conflicting Repeated Input restates, as the corresponding general principle drawn from One Input to Multiple Outputs, that repetition among input values is only ever permissible when every occurrence of that repeated input is paired with the same output, matching Same Input with the Same Output, and becomes a genuine conflict, triggering Nonfunction Classification, the moment any occurrence of that repeated input is paired with a differing output value.