Why Use a Mapping Diagram?
A relation can be described by listing pairs, but a list of pairs can make it difficult to see how each input is assigned. A mapping diagram makes these assignments visible: it places the possible inputs in one set, the possible outputs in another set, and draws an arrow for each assigned input-output pair.
The key question is not how many arrows arrive at an output. To decide whether a diagram represents a function, inspect each input and count the arrows that leave it. Every input must have exactly one outgoing arrow.
Reading the Two Sets and the Arrows
The set on the left is the source set: its elements are the inputs being assigned. The set on the right is the target set: its elements are the permitted destinations for arrows. An arrow records one assignment. For instance, an arrow from \(a\) to \(b\) records that the pair \((a,b)\) belongs to the relation being represented.
To read a diagram, select an element on the left and trace its arrow to the right. Do this separately for every source element. A diagram is not a function if even one source element has no outgoing arrow or has two or more outgoing arrows. Having exactly one outgoing arrow for each source is the function rule from the earlier material on relations and the function rule, shown in a visual form.
The arrow count is always taken at the source end. Several arrows may arrive at the same target element without causing a problem. A target element may also have no arrows arriving. Neither situation violates the function rule, because the rule requires exactly one assignment for each input, not exactly one input for each output.
A Systematic Test
Use this short procedure whenever you are given a mapping diagram. Check that each drawn arrow starts at an element of the source set and ends at an element of the target set. Then count outgoing arrows separately for each source element. The diagram defines a function if and only if every count is one.
List the elements from which arrows must leave.
Each arrow must start in the source set and end in the target set.
For each source element, count the arrows that start there, not the arrows that end there.
If every source element has exactly one outgoing arrow, the diagram defines a function. Otherwise, it does not.
Worked Examples
Worked Example: A Function with One Arrow from Each Input
Let the source set be \(A=\{-2,0,3\}\) and the target set be \(B=\{1,4,9\}\). Suppose the arrows are \(-2\to4\), \(0\to1\), and \(3\to9\). The diagram can be read as:
Count the arrows leaving each source element. The element \(-2\) has one outgoing arrow, to \(4\). The element \(0\) has one outgoing arrow, to \(1\). The element \(3\) has one outgoing arrow, to \(9\). Each endpoint belongs to \(B\), so all arrows go to permitted target elements.
Thus every element of \(A\) has exactly one outgoing arrow, and every arrow ends in \(B\). The diagram defines a function from \(A\) to \(B\). The assignments, written as ordered pairs, are \(\{(-2,4),(0,1),(3,9)\}\). Checking those pairs gives the same result: each first coordinate appears exactly once.
Worked Example: Two Arrows from One Input
Let \(A=\{a,b,c\}\) and \(B=\{1,2,3\}\). Suppose the arrows are \(a\to1\), \(a\to2\), \(b\to2\), and \(c\to3\). Count the arrows leaving each input:
- \(a\) has two outgoing arrows, one to \(1\) and one to \(2\).
- \(b\) has one outgoing arrow, to \(2\).
- \(c\) has one outgoing arrow, to \(3\).
The arrow endpoints all belong to \(B\), but \(a\) has two different assigned outputs. The requirement is exactly one outgoing arrow for every source element, so this diagram does not define a function from \(A\) to \(B\). In ordered-pair form, the relation is \(\{(a,1),(a,2),(b,2),(c,3)\}\). The repeated first coordinate \(a\), paired with two different second coordinates, displays the same failure.
Worked Example: Several Inputs Share an Output
Let \(A=\{1,2,3,4\}\) and \(B=\{\text{red},\text{blue},\text{green}\}\). Suppose the arrows are \(1\to\text{red}\), \(2\to\text{blue}\), \(3\to\text{blue}\), and \(4\to\text{blue}\). The assignments are:
Input \(1\) has one outgoing arrow. Input \(2\) has one. Input \(3\) has one. Input \(4\) has one. Thus all four source elements have exactly one assigned output. The fact that three arrows arrive at \(\text{blue}\) does not break the function rule: the rule counts arrows leaving inputs, not arrows arriving at outputs.
The element \(\text{green}\) has no incoming arrow. That is allowed; the function rule does not require every target element to be used. Therefore this diagram defines a function from \(A\) to \(B\). Its ordered pairs are \(\{(1,\text{red}),(2,\text{blue}),(3,\text{blue}),(4,\text{blue})\}\), with one pair for each element of \(A\).
Worked Example: An Input with No Arrow
Let \(A=\{p,q,r\}\) and \(B=\{x,y\}\). Suppose the diagram has arrows \(p\to x\) and \(r\to y\), but no arrow leaving \(q\). Count the outgoing arrows: \(p\) has one, \(q\) has zero, and \(r\) has one. The endpoints \(x\) and \(y\) are in the target set, but that does not fix the missing assignment for \(q\).
Because \(q\) has no outgoing arrow, it has no assigned output. The diagram does not define a function from \(A\) to \(B\). In ordered-pair form the relation is \(\{(p,x),(r,y)\}\); there is no pair whose first coordinate is \(q\). This check is complete because the function condition applies to every source element, including one that might otherwise be overlooked.
Why the Arrow Test Is Exactly the Function Rule
A relation from \(A\) to \(B\) is a set of ordered pairs whose first entries belong to \(A\) and whose second entries belong to \(B\). A mapping diagram represents each such pair by an arrow from its first entry to its second. The definition of a function requires every input in \(A\) to be paired with exactly one output in \(B\).
That definition gives both parts of the arrow test. If an input has no outgoing arrow, it appears as the first entry in no ordered pair, so it has no assigned output. If it has two or more outgoing arrows, it is paired with more than one output. Either situation fails the function requirement. Conversely, if each input has exactly one outgoing arrow and each arrow ends in \(B\), then each input is paired with exactly one permitted output. The relation therefore satisfies the definition of a function.
Common Mistake: Counting Arrows at the Wrong End
A frequent error is to require each target element to receive exactly one arrow. For example, in the diagram with \(1\to\text{red}\), \(2\to\text{blue}\), \(3\to\text{blue}\), and \(4\to\text{blue}\), someone might reason: “Blue receives three arrows, so this is not a function.” That calculation counts incoming arrows at the target, which is not the function test.
The correction is to count from the source side: \(1\) has one outgoing arrow, \(2\) has one, \(3\) has one, and \(4\) has one. The diagram is a function. A second related error is to reject a diagram because a target element receives no arrows. In the same example, \(\text{green}\) is unused, but no input is left without an assignment. Unused target elements are allowed.
Check Your Understanding
For each question, inspect the arrows leaving the source elements before deciding whether the diagram defines a function.
- Let \(A=\{u,v\}\) and \(B=\{1,2\}\), with arrows \(u\to1\) and \(v\to1\). Does the diagram define a function? Explain why.
- Let \(A=\{1,2,3\}\) and \(B=\{a,b\}\), with arrows \(1\to a\), \(2\to b\), \(2\to a\), and \(3\to b\). Which input determines whether the diagram is a function?
- If every input has exactly one outgoing arrow, but one target element has no incoming arrows, does that alone prevent the diagram from being a function?
- What is wrong with the claim, “A mapping diagram is not a function whenever two arrows arrive at the same target”?
- For \(A=\{m,n\}\), a diagram contains only the arrow \(m\to x\). What outgoing-arrow count does \(n\) have, and what does that imply?
