Into and Onto Function

Definition

Let $f: A \to B$ be a function from set $A$ to set $B$ .

Into function

: A function $f: A \to B$ is called an into function if there exists at least one element in $B$ that is not the image of any element of $A$ . In other words, the range of $f$ is a proper subset of $B$ , so $f(A) \subsetneq B$

Onto function

: A function $f: A \to B$ is called an onto function or surjective function if every element of $B$ is the image of at least one element of $A$ . In this case, $f(A) = B$

Here:

Domain

= set of all input values

Codomain

= set of all possible output values

Range/Image

= set of all actual output values produced by the function

Main Content

1. Into Function

Meaning and nature

: An into function does not use up all elements of the codomain. At least one element in the codomain remains unmatched, meaning the function does not “cover” the entire target set.

Example

: Consider $f: \{1,2,3\} \to \{a,b,c,d\}$ defined by $f(1)=a,\ f(2)=b,\ f(3)=c$ .
Here, $d$ has no preimage, so the function is into. The range is $\{a,b,c\}$ , which is a proper subset of $\{a,b,c,d\}$ .

Key idea: every function whose range is smaller than its codomain is into.

2. Onto Function

Meaning and nature

: An onto function covers the entire codomain. Every element of the codomain has at least one corresponding element in the domain.

Example

: Let $g: \{1,2,3\} \to \{a,b,c\}$ be defined by $g(1)=a,\ g(2)=b,\ g(3)=c$ .
Since each element of $\{a,b,c\}$ is hit by the function, $g$ is onto.

Many-to-one allowed

: An onto function does not require different domain elements to map to different codomain elements. For example, if $h: \{1,2,3\} \to \{a,b\}$ with $h(1)=a,\ h(2)=a,\ h(3)=b$ , then $h$ is onto even though two inputs share the same output.

Key idea: onto means “every element in the codomain is reached.”

3. Difference Between Into and Onto

Coverage of codomain

:
Into: not all codomain elements are used
Onto: all codomain elements are used

Range relation

:
Into: range is a proper subset of codomain
Onto: range equals codomain

Practical interpretation

:
Into functions leave “unused outputs”
Onto functions ensure complete coverage of outputs

Visual understanding

For the function $f: A \to B$ :

If $B$ has elements not pointed to by arrows, it is into
If every element of $B$ has at least one arrow pointing to it, it is onto

A = {1, 2, 3}                 B = {a, b, c, d}

1  ─────► a
2  ─────► b
3  ─────► c

d is not reached, so the function is into.

A = {1, 2, 3}                 B = {a, b, c}

1  ─────► a
2  ─────► b
3  ─────► c

Every element of B is reached, so the function is onto.

Working / Process

1. Identify the domain and codomain

Write down the sets $A$ and $B$ clearly.
Do not confuse codomain with range; codomain is the declared target set.

2. Find the images of all domain elements

Apply the function rule to each element of the domain.
Collect all outputs to form the range.

3. Compare range with codomain

If every element of the codomain appears in the range, the function is onto.
If at least one element of the codomain is missing, the function is into.
If needed, verify by checking whether each codomain element has a preimage.

Advantages / Applications

Classification of functions

: Helps determine the exact nature of a function, especially whether it covers all possible outputs.

Foundation for inverse functions

: Onto functions are important when studying inverse mappings; a function must be onto to have an inverse on the codomain, provided it is also one-to-one.

Use in mathematics and computing

: These concepts are used in algebra, abstract algebra, discrete mathematics, database mappings, algorithm design, and theorem proving to understand completeness, representation, and mapping behavior.

Summary

An into function leaves at least one element of the codomain unmapped.
An onto function maps onto every element of the codomain.
The main test is comparing the range with the codomain.