Bijective Function in Automata Theory

---

---

A bijective function, also known as a bijection, is a function that is both one-to-one (injective) and onto (surjective). These concepts are important in mathematics, particularly in set theory and other branches that involve functions. We will see the basic definitions, provide examples, and explain the criteria for a function to be bijective for a better understanding.

What is a Bijective Function?

A bijective function is a function that satisfies two main criteria −

• One-to-One (Injective) − Every element in the domain (Set A) maps to a distinct element in the codomain (Set B).
• Onto (Surjective) − Every element in the codomain (Set B) is mapped by at least one element in the domain (Set A).

If a function meets both these criteria, it is called a bijection.

Formally we can say, a function f from Set A to Set B is bijective if it is both one to one and onto. This means that each element in Set A maps to a unique element in Set B, and every element in Set B has a corresponding element in Set A.

In simpler terms, in a bijective function, there is a perfect pairing between the elements of Set A and Set B.

Understanding One-to-One (Injective) Function

Let us recap the one-to-one function which is needed for bijective functions.

One-to-One Mapping

A function is one-to-one if each element in the domain (Set A) has a unique image in the codomain (Set B). In other words, no two different elements in Set A map to the same element in Set B.

For example −

$$\mathrm{Set \:A \:=\: \{1,\: 2,\: 3\}}$$

$$\mathrm{Set \:B \:=\: \{a,\: b,\: c,\: d\}}$$

If we map 1 to a, 2 to b, and 3 to c, the function is one-to-one because each element in Set A has a distinct image in Set B.

Example of Non-One-to-One Mapping

Suppose Set A has four elements {1, 2, 3, 4} and Set B has three elements {a, b, c}, and we map as shown in the figure below. This function is not one-to-one because two different elements in Set A (3 and 4) map to the same element in Set B (C).

The key takeaway idea is that for a function to be one to one, the number of elements in Set A must be less than or equal to the number of elements in Set B.

Understanding Surjective (Onto) Function

Let's recap the concept of onto functions.

Onto Mapping

A function is onto if every element in the codomain (Set B) is mapped by at least one element in the domain (Set A). In an onto function, no element in Set B is left unmapped.

For example −

$$\mathrm{Set \:A \:=\: \{1,\: 2,\: 3,\: 4\}}$$

$$\mathrm{Set \:B \:=\: \{a,\: b,\: c\}}$$

Example of Non-Onto Mapping

If $\mathrm{Set \:A \:=\: \{1,\: 2,\: 3\}}$ and $\mathrm{Set \:B \:=\: \{a,\: b,\: c,\: d\}}$, and we map −

Element d in Set B is left unmapped, so the function is not onto.

The criterion for a function to be onto is that the number of elements in Set A should be greater than or equal to the number of elements in Set B. If Set A has fewer elements than Set B, the function cannot be onto.

Combining One-to-One and Onto: The Bijective Function

A function is bijective if it is both one-to-one and onto. For a function to be bijective, the number of elements in Set A must be equal to the number of elements in Set B.

Example of a Bijective Function

Let's consider an example where Set A and Set B have the same number of elements −

$$\mathrm{Set \:A \:=\: \{1,\: 2,\: 3,\: 4\}}$$

$$\mathrm{Set \:B \:=\: \{a,\: b,\: c,\: d\}}$$

We can map −

This function is both one-to-one (each element in Set A has a distinct image) and onto (each element in Set B has a pre-image). Therefore, it is a bijection.

Example of Non-Bijective Function

Consider the function −

$$\mathrm{Set \:A \:=\: \{1,\: 2,\: 3,\: 4\}}$$

$$\mathrm{Set \:B \:=\: \{A,\: B,\: C\}}$$

If we map −

This function is neither one-to-one (because 3 and 4 map to the same element in Set B) nor onto (because not all elements in Set B have pre-images). Therefore, it is not a bijection.

Bijective Function with Infinite Sets

The concept of bijective functions becomes more useful when used in infinite sets. For finite sets, if Set A and Set B have the same number of elements, and the function is one-to-one, it is also onto, making it bijective. However, this is not always the case with infinite sets.

Example of Bijective Function with Infinite Sets

Let us consider a function from the set of natural numbers to itself −

$$\mathrm{Set \:A \:=\: \{1,\: 2,\: 3,\: \dotso\}}$$

$$\mathrm{Set \:B \:=\: \{1,\: 2,\: 3,\: \dotso\}}$$

If we define the function f(x) = x, where each natural number maps to itself, the function is both one-to-one and onto. Therefore, it is a bijection.

Example of Non-Bijective Function with Infinite Sets

Now, consider a different function −

$$\mathrm{Set \:A \:=\: \{1,\: 2,\: 3,\: \dotso\}}$$

$$\mathrm{Set \:B \:=\: \{1,\: 2,\: 3,\: \dotso\}}$$

If we define the function f(x) = 2x, where −

This function is one-to-one, but not onto because odd numbers in Set B have no pre-images in Set A. Therefore, it is not a bijection.

Counting the Number of Bijective Function

The number of bijective functions between two finite sets with the same number of elements can be calculated using factorials.

Example: Calculating Number of Bijections

If Set A and Set B both have n elements, then the number of bijective functions is n! (n factorial).

For example, if −

$$\mathrm{Set \:A \:=\: \{1,\: 2,\: 3\}}$$

$$\mathrm{Set \:B \:=\: \{A,\: B,\: C\}}$$

The number of bijective functions is −

$$\mathrm{3! \:=\: 3 \:\times\: 2 \:\times\: 1 \:=\: 6}$$

It means, there are six possible bijections between these two sets.

Conclusion

In this chapter, we explained the concept of bijective functions. We discussed examples of both bijective and non-bijective functions, including cases with finite and infinite sets. We also learned how to calculate the number of bijective functions between two finite sets.