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Found in: Page 439

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# a) How can the product rule be used to find the number of functions from a set with m elements to a set with n elements?b) How many functions are there from a set with five elements to a set with 10 elements?c) How can the product rule be used to find the number of one-to-one functions from a set with m elements to a set with n elements?d) How many one-to-one functions are there from a set with five elements to a set with 10 elements?e) How many onto functions are there from a set with five elements to a set with 10 elements?

(a) The number of function from a set with m elements to a set with n element $={\mathrm{n}}^{\mathrm{m}}$ .

(b) The number of functions from a set having 5 elements to a set having 10 elements $={10}^{5}$ .

(c) Total number of one to one function from a set with m elements to a set with n elements $=\mathrm{n}\left(\mathrm{n}-1\right)\left(\mathrm{n}-2\right)\dots \dots ..\left(\mathrm{n}-\mathrm{m}+1\right)$.

(d) Total number of one to one function is 30240 .

(e) Number of onto function =0 .

See the step by step solution

## Step 1: Definition of Concept

Functions: It is a expression, rule or law which defines a relationship between one variable and another variables.

## Step 2: Find the number of function from a set with m elements to a set with n element

(a)

Considering the given information:

Number of element =m

Number of element =n

Using the following concept:

Order of lexicography

The number of elements from a set with p element to a set with q element $={\mathrm{p}}^{2}$ .

The number of elements transferred from a set with m element to a set with n element is

$=\mathrm{n}×\mathrm{n}×\dots \dots ..\mathrm{m}$

Therefore, the required number of function from a set with m elements to a set with n element $={\mathrm{n}}^{\mathrm{m}}$ ..

## Step 3: Find the number of functions from a set having 5 elements to a set having 10 elements

(b)

Considering the given information:

Number of element =5

Number of set of element =10

Using the following concept:

Number of element m to a set with $\mathrm{n}={\mathrm{n}}^{\mathrm{m}}$.

Here

n=10

m=5

Number of functions from a set having 5 elements to a set having 10 elements $={10}^{5}$.

Therefore, the required number of functions from a set having 5 elements to a set having 10 elements $={10}^{5}$ .

## Step 4: Find the number of one-to-one functions from a set with m elements to a set with n elements

(c)

Considering the given information:

Number of element =m

Number of set of element =n

Using the following concept:

For a single input, a one to one function returns a single value.

Number of element of one to one function from a set with m elements to a set with n elements is $\mathrm{n}\left(\mathrm{n}-1\right)\left(\mathrm{n}-2\right)\dots \dots ..\left(\mathrm{n}-...+1\right)$

If n=m

Total number $=\mathrm{n}\left(\mathrm{n}-1\right)\left(\mathrm{n}-2\right)\dots \dots \dots ..1$

=n!

Therefore, the required total numbers are role="math" localid="1668680508265" $\mathrm{n}\left(\mathrm{n}-1\right)\left(\mathrm{n}-2\right)\dots \dots ..\left(\mathrm{n}-\mathrm{m}+1\right)$ .

## Step 5: Find the total number of one to one function

(d)

Considering the given information:

Number of set of element =5

Number of element in larger set = 10

Using the following concept:

Number of one to one functions from a set having m elements to a set having n element

$=\mathrm{n}\left(\mathrm{n}-1\right)\left(\mathrm{n}-2\right)\dots \dots \dots \dots ..\left(\mathrm{n}-\mathrm{m}+1\right)$

Here,

m=5

n=10

As a result, the number of one-to-one functions,

$=10\left(10-1\right)\left(10-2\right)\left(10-3\right)\left(10-4\right)\phantom{\rule{0ex}{0ex}}=10×9×8×7×6\phantom{\rule{0ex}{0ex}}=30240$

Therefore, the required total number of one to one functions is 30240 .

## Step 6: Find the number of onto function

(e)

Considering the given information:

Number of set of element =5

Number of larger set =10

Using the following concept:

From a set with fewer elements to a set with more elements, the onto function cannot be defined.

Because the onto function cannot be defined from a set of fewer elements to a set of more elements.

As a result, the number of onto functions has increased from five to ten =0.

Therefore, the required total number is 0.