Types of functions | Part -2

Into, Injective, and Bijective Functions with Illustrations

In mathematics, understanding how functions map elements from one set to another is essential. Different types of functions behave differently, and we can categorize them based on how elements in the domain are related to the codomain. This article focuses on three important types: into functions, injective (one-one) functions, and bijective functions, with explanations and examples using arrow diagrams.

1. Into Function

A function is said to be an into function if there is at least one element in the codomain that is not the image of any element in the domain. In simple terms, some codomain values are left unused.

Example:
Let domain A = {x, y, z} and codomain B = {a, b, c, d}.
Define function f = {(x, a), (y, b), (z, c)}.
Here, 'd' in the codomain is not mapped to by any element in the domain, so this is an into function.

Visual Clue: In an arrow diagram, one or more codomain elements have no arrows pointing to them.

2. Injective Function (One-One)

A one-one function, or injective function, is defined such that each element in the domain maps to a distinct element in the codomain. No two inputs share the same output.

Example:
Let domain A = {x, y, z} and codomain B = {a, b, c, d}.
Define function f = {(x, a), (y, b), (z, d)}.
All outputs are distinct, so no two inputs share the same image. Hence, this is an injective function.

Visual Clue: In an arrow diagram, each codomain element has at most one arrow pointing to it.

Note: An injective function may or may not be onto. If it's not onto, it's simply one-one into.

3. How to Identify a Bijective Function

A function is called bijective if it is both injective (one-one) and surjective (onto). In other words:

• Every element in the domain is assigned to a distinct element in the codomain (injective).
• Every element in the codomain is covered (surjective).

Example of a Bijective Function:

Let A = {x, y, z} and B = {a, b, c}
f = {(x, a), (y, b), (z, c)}
This is bijective because all outputs are unique (one-one) and every codomain element is mapped (onto).

Non-Bijective Case 1 – Not Onto:

If B = {a, b, c, d} and f = {(x, a), (y, b), (z, c)}
The element 'd' is not mapped, so the function is not onto, and therefore not bijective.

Non-Bijective Case 2 – Not One-One:

If f = {(x, a), (y, a), (z, b)}
Both x and y map to 'a', so the function is not injective, and hence not bijective.

Checking Whether a Function is Onto (Surjective)

To check if a function is onto, ensure that each element in the codomain is mapped by one or more elements from the domain. In other words, there should be no element in the codomain that remains unmatched.

Here’s how to test it:

1. List the codomain elements: Clearly identify all elements in the codomain.
2. Trace all mappings: Check which codomain elements are mapped by elements from the domain.
3. Compare: If all codomain elements are covered (i.e., have at least one arrow pointing to them in an arrow diagram or a corresponding pre-image), the function is onto.

Example: Let ( f_3: X = {x, y, z} rightarrow Y = {a, b, c} ) with mappings: ( f_3(x) = a, f_3(y) = b, f_3(z) = c ). Since each element in Y is mapped by some element in X, this is an onto function.

If even one codomain element has no pre-image, the function is not onto.

In summary, identifying bijective functions involves checking for both uniqueness (injectivity) and completeness (surjectivity). Arrow diagrams are helpful tools for quickly analyzing these properties.