Once you have a diagram for each situation, look at the different features of each diagram.
What do you notice?
Are there similarities/differences between the situations?
Can any of them be grouped together because they have certain features?
If we take the first diagram from our car example (where ‘multi-coloured’ was a separate category), what features do we notice that might appear elsewhere?
All the inputs (cars) have an output (colour).
Different inputs can go to the same output.
The same input cannot go to more than one output.
Can we classify all the situations using these features?
Remember that there will be different ways to draw diagrams for some of these situations depending on how you thought about them. What is important is not necessarily having the same diagrams, but thinking about the features in a meaningful way.
One-to-one
All the inputs go to a single, unique output.
Situation \(6\) has every input (a person) going to its own output (thumb print).
One-to-many
An input can go to several (or many) different outputs.
In situation \(1\), if you know your height above sea level (input), it is likely there will many different places where you could be on a map (output).
Many-to-one
Different inputs can go to the same output as each other.
In situation \(3\), lots of people who voted (input) will have voted for the same person (output). Situation \(5\) is similar.
Many-to-many
Different inputs can go to the same output as each other and inputs can go to more than one output.
It is likely that between your friends, some of you follow the same people, and very likely that, if you are on Twitter, you follow more than one person.
If you follow one of your friends, you would not connect two dots in the first oval. The input is the group of friends, the output is who they follow, so if you were following a friend, they would appear on both sides of the diagram as an input, and an output. Of course if none of you are on Twitter then your diagram would not look like this at all!
What’s left?
The only situation we haven’t looked at is \(2\). What is different about this diagram?
Even though people are eligible to vote, they may not have done so. So we have inputs that don’t go to any output.
What is different between the inputs of situation \(2\) and \(3\)?
These situations and diagrams all describe the relationships between the inputs and outputs. You have probably met the idea of a function before, which has inputs and outputs. If we define a function as a relationship where every input has exactly one output, which of the situations above are functions?
Let us return to our example, and the two options we had for the diagrams.
Cars map to one colour only, and no cars are left out. (Includes a ‘multi-coloured’ category)
Cars map to more than one colour. (No ‘multi-coloured’ category)
We are looking for situations, where for a given input, we will have only one possible output. Clearly the right-hand diagram does not offer this, as some cars go to more than one colour. The left-hand diagram is a function, as each car goes to only one colour.
From our groupings above, a one-to-one and many-to-one relationship both describe functions.
Our definition of a function is that an input must have exactly one output, so situation \(3\) does not count, as there are inputs which don’t have any output. When dealing with mathematical functions, it is important that the set of inputs is carefully considered to ensure this part of the definition of a function is met.
