Mapping Diagrams Explained: Complete Guide with Examples (2026)

Mapping diagrams are one of the most intuitive ways to visualize relationships between two sets of values. In mathematics, they make it immediately clear whether a relation is a function, what type of function it is, and how elements in one set connect to elements in another.

This guide explains everything you need to know about mapping diagrams: what they are, how to create them, the different types of mappings they represent, how they compare to other function representations, and where they appear in both mathematics education and research.

What Is a Mapping Diagram?

A mapping diagram (also called an arrow diagram or transformation figure) is a visual representation that shows the relationship between two sets by drawing arrows from elements in one set to elements in another. It consists of:

• Domain -- the set of input values (also called Set A or the source set), usually drawn on the left
• Codomain -- the set of possible output values (also called Set B or the target set), usually drawn on the right
• Arrows -- lines connecting each input to its corresponding output, showing how the mapping works

Here is a simple example of a mapping diagram:

  Domain          Codomain
 ┌───────┐      ┌───────┐
 │   1  ─┼──────┼→  a   │
 │   2  ─┼──────┼→  b   │
 │   3  ─┼──────┼→  c   │
 └───────┘      └───────┘

In this diagram, the function maps 1 to a, 2 to b, and 3 to c. Each element in the domain is paired with exactly one element in the codomain.

Key Terminology

Mapping Diagrams vs. Functions

A mapping diagram represents a relation between two sets. Not every relation is a function. A relation qualifies as a function only when each input maps to exactly one output. In a mapping diagram, this means:

• Function: Every element in the domain has exactly one arrow going out
• Not a function: At least one element in the domain has two or more arrows going to different outputs

  FUNCTION               NOT A FUNCTION
 ┌───────┐ ┌───────┐    ┌───────┐ ┌───────┐
 │   1  ─┼→┼  5    │    │   1  ─┼→┼  5    │
 │   2  ─┼→┼  10   │    │   2  ─┼→┼  10   │
 │   3  ─┼→┼  15   │    │   3  ─┼→┼  15   │
 └───────┘ └───────┘    │      ─┼→┼  20   │
                         └───────┘ └───────┘
                         (3 maps to both 15 AND 20)

Mapping diagrams provide a discrete, visual way to represent the same relationships shown by function graphs -- making them especially useful for checking whether a relation is actually a function

Types of Mappings

Understanding the different types of mappings is essential for working with functions in algebra, set theory, and higher mathematics.

One-to-One (Injective) Mapping

In a one-to-one mapping, each element in the codomain is mapped to by at most one element in the domain. No two different inputs produce the same output.

  ONE-TO-ONE (INJECTIVE)
 ┌───────┐      ┌───────┐
 │   1  ─┼──────┼→  a   │
 │   2  ─┼──────┼→  b   │
 │   3  ─┼──────┼→  c   │
 │        │      │   d   │  ← no arrow points here
 └───────┘      └───────┘

How to check on a mapping diagram: No element in the codomain has more than one arrow pointing to it. Each arrow lands on a unique element.

Real-world example: Social Security numbers -- each person has a unique number, and no two people share the same number.

Onto (Surjective) Mapping

In an onto mapping, every element in the codomain is mapped to by at least one element in the domain. No element in the codomain is left without an arrow.

  ONTO (SURJECTIVE)
 ┌───────┐      ┌───────┐
 │   1  ─┼──────┼→  a   │
 │   2  ─┼──────┼→  b   │
 │   3  ─┼──┐   │       │
 │   4  ─┼──┘───┼→  c   │  ← two arrows point here
 └───────┘      └───────┘

How to check on a mapping diagram: Every element in the codomain has at least one arrow pointing to it. The range equals the codomain.

Real-world example: A seating chart where every seat must be filled -- multiple students might share a table, but no table is empty.

Bijective (One-to-One and Onto) Mapping

A bijective mapping is both one-to-one and onto. Every element in the domain maps to a unique element in the codomain, and every element in the codomain is mapped to. This creates a perfect "one-to-one correspondence."

  BIJECTIVE (ONE-TO-ONE AND ONTO)
 ┌───────┐      ┌───────┐
 │   1  ─┼──────┼→  a   │
 │   2  ─┼──────┼→  b   │
 │   3  ─┼──────┼→  c   │
 └───────┘      └───────┘

Key property: Bijective functions always have an inverse function. You can reverse all the arrows and get a valid function.

How to check on a mapping diagram: The domain and codomain have the same number of elements, each element in the domain has exactly one arrow, and each element in the codomain receives exactly one arrow.

Many-to-One Mapping

In a many-to-one mapping, two or more elements in the domain map to the same element in the codomain. This is still a valid function (each input has one output), but it is not injective.

  MANY-TO-ONE
 ┌───────┐      ┌───────┐
 │   1  ─┼──┐   │       │
 │  -1  ─┼──┘───┼→  1   │
 │   2  ─┼──┐   │       │
 │  -2  ─┼──┘───┼→  4   │
 └───────┘      └───────┘

Example: The function f(x) = x² maps both 2 and -2 to 4. This is a function, but not one-to-one.

Summary of Mapping Types

How to Create a Mapping Diagram Step by Step

Step 1: Identify the Domain and Codomain

Define the two sets you are working with. The domain contains your input values, and the codomain contains the possible output values.

Example: For the function f(x) = 2x, using the domain {1, 2, 3, 4}:

• Domain: {1, 2, 3, 4}
• Codomain: {2, 4, 6, 8}

Step 2: Draw Two Ovals or Columns

Draw two enclosed shapes (ovals, rectangles, or columns) side by side. Label the left shape "Domain" and the right shape "Codomain" (or "Range").

Step 3: List the Elements

Write each element of the domain inside the left shape and each element of the codomain inside the right shape. List each value only once, even if it appears multiple times in your data.

Step 4: Draw the Arrows

For each input value, draw an arrow from it to its corresponding output value. Follow these rules:

• Every element in the domain must have at least one arrow going out (for a total function)
• For a function, each element in the domain must have exactly one arrow going out
• Arrows can cross if necessary--crossing arrows do not change the meaning

Step 5: Verify the Mapping

Check your completed diagram:

• Is it a function? Each input has exactly one arrow.
• Is it one-to-one? No output has more than one arrow pointing to it.
• Is it onto? Every output has at least one arrow pointing to it.
• Is it bijective? Both one-to-one and onto.

Worked Example

Create a mapping diagram for f(x) = x² + 1 with domain {-2, -1, 0, 1, 2}:

1. Calculate outputs:

   • f(-2) = 4 + 1 = 5
   • f(-1) = 1 + 1 = 2
   • f(0) = 0 + 1 = 1
   • f(1) = 1 + 1 = 2
   • f(2) = 4 + 1 = 5

2. Draw the mapping diagram:

  Domain          Codomain
 ┌───────┐      ┌───────┐
 │  -2  ─┼──┐   │       │
 │   2  ─┼──┘───┼→  5   │
 │  -1  ─┼──┐   │       │
 │   1  ─┼──┘───┼→  2   │
 │   0  ─┼──────┼→  1   │
 └───────┘      └───────┘

3. Analysis:
   • Is it a function? Yes--each input has exactly one output
   • Is it one-to-one? No--both -2 and 2 map to 5, and both -1 and 1 map to 2
   • Is it onto? Yes--every element in the codomain {1, 2, 5} has at least one arrow
   • Type: Many-to-one, surjective (onto)

Mapping Diagrams vs. Other Representations

In mathematics, relationships between inputs and outputs can be shown in several ways. Each representation has distinct strengths. Here is how mapping diagrams compare.

Mapping Diagrams vs. Tables

Example comparison:

In the mapping diagram, the output 5 is listed only once with two arrows pointing to it, making the many-to-one nature immediately visible.

Mapping Diagrams vs. Graphs

Mapping Diagrams vs. Equations

When to Use Each Representation

While graphs excel at showing continuous relationships and trends, mapping diagrams are better suited for visualizing discrete input-output pairs and verifying function properties

Applications in Mathematics Education

Mapping diagrams are widely used in algebra and pre-algebra courses to introduce students to the concept of functions. They serve several pedagogical purposes.

Teaching Functions (Algebra 1 and Pre-Algebra)

Mapping diagrams are often the first representation students encounter when learning about functions because they make the definition concrete:

• Students can physically draw arrows and see that "each input has exactly one output"
• The visual nature makes abstract concepts tangible
• Errors (like one-to-many relations) are immediately visible

Common Classroom Activities

1. Function or not? -- Students examine mapping diagrams and determine which represent functions
2. Create from tables -- Students convert tables of values into mapping diagrams
3. Create from equations -- Students evaluate a function at specific inputs and draw the resulting mapping
4. Identify the type -- Students classify mappings as injective, surjective, or bijective
5. Real-world mappings -- Students create mapping diagrams for everyday relationships (students to grades, items to prices)

Mapping Diagrams in Set Theory

In set theory courses, mapping diagrams illustrate:

• Relations vs. functions -- not all relations are functions, but all functions are relations
• Composition of functions -- chaining two mapping diagrams shows how g(f(x)) works
• Inverse functions -- reversing all arrows in a bijective mapping gives the inverse
• Cardinality -- bijective mappings prove two sets have the same number of elements

Mapping Diagrams in Calculus Preparation

Before studying limits and continuity, students benefit from understanding:

• Domain restrictions -- which inputs are valid for a function
• Range determination -- which outputs a function can produce
• Function behavior -- one-to-one functions pass the horizontal line test

Applications in Research

Beyond mathematics education, mapping diagrams and the concept of mappings appear in several research contexts.

Data Mapping in Database Design

Data mapping defines how fields in one dataset correspond to fields in another. Researchers use mapping diagrams (or similar visuals) to:

• Map survey questions to database columns -- ensuring every response is captured correctly
• Transform data between formats -- defining how CSV fields map to database tables
• Integrate datasets -- showing how variables in one study correspond to variables in another

For more on database design, see our ER diagram complete guide for research.

Concept Mapping

While distinct from mathematical mapping diagrams, concept maps share the idea of connecting elements from different categories:

• Theoretical frameworks -- mapping variables to constructs
• Literature reviews -- mapping findings to research questions
• Taxonomy development -- mapping specimens to classifications

Category Theory

In advanced mathematics and theoretical computer science, mapping diagrams (called commutative diagrams) are fundamental to category theory. They show how objects and morphisms relate, and whether different paths through the diagram produce the same result.

Common Mistakes When Creating Mapping Diagrams

Mistake 1: Listing Values More Than Once

Problem: Writing the same output value multiple times when multiple inputs map to it.

Fix: List each value only once in the domain or codomain. Use multiple arrows pointing to the same value.

  WRONG                     RIGHT
 ┌───────┐ ┌───────┐      ┌───────┐ ┌───────┐
 │   1  ─┼→┼  5    │      │   1  ─┼─┐       │
 │   3  ─┼→┼  5    │      │   3  ─┼─┘→  5   │
 └───────┘ └───────┘      └───────┘ └───────┘
 (5 listed twice)          (5 listed once)

Mistake 2: Confusing Codomain with Range

Problem: Only listing output values that are actually used, rather than all possible output values.

Fix: The codomain includes all possible outputs, while the range includes only the outputs that are actually mapped to. In a mapping diagram, you may include codomain elements that have no arrows pointing to them.

Mistake 3: Forgetting to Map All Domain Elements

Problem: Leaving some input values without arrows.

Fix: In a total function, every element in the domain must have an arrow. If an input has no defined output, it means the function is not defined at that point (a partial function).

Mistake 4: Drawing Multiple Arrows from One Input

Problem: Accidentally creating a one-to-many relation, which is not a function.

Fix: Double-check that each domain element has exactly one arrow going out. If you find multiple arrows from one input, your relation is not a function.

Mistake 5: Confusing the Direction of Arrows

Problem: Drawing arrows from the codomain to the domain instead of domain to codomain.

Fix: Arrows always go from left to right (domain to codomain). This represents the direction of the mapping: input → output.

Mistake 6: Assuming Crossing Arrows Mean an Error

Problem: Thinking that arrows that cross each other indicate a mistake.

Fix: Crossing arrows are perfectly valid. They just mean that the mapping does not preserve order. For example, mapping {1, 2} to {b, a} where 1→b and 2→a produces crossing arrows, but the function is still valid.

Tools for Creating Mapping Diagrams

Manual Methods

• Pen and paper -- the fastest method for small diagrams in class or exams
• Whiteboard -- ideal for classroom demonstrations
• Graph paper -- helps maintain alignment for neat diagrams

Digital Tools

AI-Powered Diagram Creation

For researchers and students who need to create professional-quality mapping diagrams quickly, AI-powered tools offer an efficient alternative to manual drawing.

Text to Diagram Generator

Describe your mapping diagram in plain language and get a professional visual output instantly. No design skills needed.

Try it free →

You can also create related visualizations with these tools:

• Venn Diagram Generator -- for showing set relationships and overlaps
• Conceptual Framework Generator -- for mapping research variables to constructs

Mapping Diagram Practice Examples

Example 1: Determine if the Relation Is a Function

Given the mapping diagram:

 ┌───────┐      ┌───────┐
 │   2  ─┼──────┼→  4   │
 │   5  ─┼──────┼→  7   │
 │   8  ─┼──────┼→  4   │
 │  11  ─┼──────┼→  13  │
 └───────┘      └───────┘

Answer: Yes, this is a function. Each input (2, 5, 8, 11) has exactly one arrow going out. The fact that both 2 and 8 map to 4 is allowed--it makes this a many-to-one function, but it is still a valid function.

Example 2: Create a Mapping Diagram from Ordered Pairs

Given the relation: {(1, 3), (2, 6), (3, 9), (4, 12)}

 ┌───────┐      ┌───────┐
 │   1  ─┼──────┼→  3   │
 │   2  ─┼──────┼→  6   │
 │   3  ─┼──────┼→  9   │
 │   4  ─┼──────┼→  12  │
 └───────┘      └───────┘

Analysis: This is a function (each input has one output). It is one-to-one (no two inputs share an output). It represents the function f(x) = 3x.

Example 3: Identify the Mapping Type

Given the mapping diagram:

 ┌───────┐      ┌───────┐
 │   a  ─┼──────┼→  1   │
 │   b  ─┼──────┼→  2   │
 │   c  ─┼──────┼→  3   │
 └───────┘      └───────┘

Answer: This is a bijective (one-to-one and onto) function. Every input maps to a unique output, and every output is mapped to. This function has an inverse: {(1, a), (2, b), (3, c)}.

Example 4: Real-World Mapping

A teacher maps students to their test scores:

  Students       Scores
 ┌───────┐      ┌───────┐
 │  Alice─┼──────┼→  92  │
 │  Bob  ─┼──┐   │       │
 │  Carol─┼──┘───┼→  85  │
 │  David─┼──────┼→  78  │
 └───────┘      └───────┘

Analysis: This is a function (each student has one score). It is not one-to-one because Bob and Carol both scored 85. It is a many-to-one function.

Frequently Asked Questions

Conclusion

Mapping diagrams are a foundational tool for understanding functions and relations in mathematics. Their visual simplicity makes them ideal for:

1. Verifying functions -- quickly check if each input has exactly one output
2. Classifying mappings -- determine if a function is injective, surjective, or bijective
3. Teaching concepts -- make abstract function definitions concrete and visual
4. Communicating relationships -- show discrete input-output pairs clearly

Key takeaways:

• A mapping diagram is a function only if each input has exactly one arrow going out
• One-to-one functions have no shared outputs; onto functions leave no outputs unmapped
• Bijective functions are both one-to-one and onto, and always have inverses
• Mapping diagrams work best for finite, discrete sets -- use graphs for continuous functions
• List each value only once in the domain or codomain, using multiple arrows as needed

Whether you are a student learning about functions for the first time or a researcher mapping data between systems, understanding mapping diagrams gives you a clear, visual foundation for working with mathematical relationships.

Additional Resources

• Injective, Surjective and Bijective Functions
• Representing Functions with Mappings, Tables, and Graphs
• How to Create a Conceptual Framework for Research
• ER Diagram Complete Guide for Research
• Text to Diagram Generator
• Venn Diagram Generator