# Relations and Functions

A relation is an ordered set of two or more terms with functions, each having a single input and output. It can be viewed as representing the relationship between input and output, with each term being dependent on the other.

While, a relation represents a relationship that derives a single output for every single input, an independent set of terms is called a functional relation. Note: A functional relation is not a relation in general, it can only be viewed as an independent set of terms, with every term dependent on another. In general, a functional relation can be both an independent and dependent relation, it can also be a dependent and independent relation.

A function represents a relationship that has a single input, and a single output, in a non-deterministic and non-transitive way. One way of looking at this is that a function can be seen as an “incomplete” object. Another way of looking at it is that a function is just the union of all possible relations in which a term occurs. Note that a functional relation is not necessarily a relation in general, as a functional relation can be an independent and a dependent relation.

A relation is the concept of a relationship between a set of terms and their corresponding outputs. For example, a relation between cars and taxis can be described using the following diagram. On the left side of the diagram, we see cars and taxis. On the right side of the diagram, we see different outputs of the two terms. This diagram can also be considered as describing the relation between taxis and cars.

The input of the first term, the car, is associated with the state of the driver in terms of either the car’s speed or the traffic light at the time the car was stopped. The output of the second term, the taxi, is associated with the actual arrival time of the taxi. Since both are dependent on the state of the driver at the time of the first occurrence of the first term, the relation is called a “proxemic” relation.

A relation between terms is called a “monotonic” relation, if there is only one output associated with the relation. The output of the first term, the taxi, is not changed by the arrival of a second term, the car, in terms of either the state of the driver at the time of the first occurrence of the first term, or the second term, the traffic light.

A relation between terms is called a “diagonal” relation if there are two inputs and two outputs. The input of the first term, the car, is related to the output of the second term, the traffic light, by the state of the driver. The output of the second term, the car, is related to the first term, the taxi, by the availability of a third term, the available space of cars in the immediate vicinity of the car in terms of both the availability of space and of light at the time of the first occurrence of the first term, the car.

A relation between terms is called a “diagonal” relation, if there is more than one output associated with the relation. The output of the first term, the car, is related to the second term, the traffic light, by the availability of a third term, the availability of light and space, at the time of the first occurrence of the first term, the car and the availability of cars in the immediate vicinity of the first term, the traffic light.