Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

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 $= n^{m}$ .

(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 $= n (n - 1) (n - 2) \dots \dots . . (n - m + 1)$ .

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

(e) Number of onto function =0 .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 $= p^{2}$ .

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

$= n \times n \times \dots \dots . . m$

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

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 $n = n^{m}$ .

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 $n (n - 1) (n - 2) \dots \dots . . (n - . . . + 1)$

If n=m

Total number $= n (n - 1) (n - 2) \dots \dots \dots . . 1$

=n!

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

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

$= n (n - 1) (n - 2) \dots \dots \dots \dots . . (n - m + 1)$

Here,

m=5

n=10

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

$= 10 (10 - 1) (10 - 2) (10 - 3) (10 - 4) = 10 \times 9 \times 8 \times 7 \times 6 = 30240$

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.

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Discrete Mathematics and its Applications

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Short Answer

Step by Step Solution

Step 1: Definition of Concept

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

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

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

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

Step 6: Find the number of onto function

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Discrete Mathematics and its Applications

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Short Answer

Step by Step Solution

Step 1: Definition of Concept

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

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

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

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

Step 6: Find the number of onto function

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