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# Onto Functions

If A and B are the two sets, we call it the onto function if, for every element of Y, there are at least one or more elements that match with set X.

The surjective function is another name for the onto function.

It is a function f that maps any element x to every element y. There is an x such that f(x) = y for every y. Every element of the function's co-domain is an image of at least one element of the function's domain.

If at least one or more elements is matching with X for every element of Y, we say that the function is an onto function or surjective function. However, one element in Set Y is not mapped to any element in Set X, showing that it is not an onto or surjective function.

## Examples of onto function:

If a number is divisible by two, then that respective number is an even number; otherwise, it is an odd number. Thus, when we divide a number by two, the remainder will be an odd number.

## What is the distinction between the onto and into functions?

When we represent a function using Venn diagrams, we call it mapping.

We define mappings between sets X and Y such that Y has at least one element, y, that is not the f-image of X. If every element of Y is the f-image of at least one element of X; we say that the mapping of f to be onto.

## How do you determine whether a function is onto?

Codomain = Range

A function's co-domain is the set of its potential outputs. For example, the function notation f:RðŸ¡ªR denotes f is a function from real numbers to real numbers.

If it corresponds to the values of the co-domain, then the function is onto in that domain. When the function's range equals the function's co-domain, we say the function to be onto or surjective, and when the range is completely a subset of the co-domain, we say that the function is into.

## How do we prove that f(x) is an onto function?

If and only if all elements in B can find some elements in A with the property that y = f(x), where y is B and x is A, then f is onto y B, x A such that f(x) = y.

## How many functions are onto?

The total number of functions from a set of m elements to a set of 2 elements is 2m. Two of these functions are not on the list. If we map all elements to the 1st element of Y or the 2ndelement of Y, the number of onto functions would be 2m-2.

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