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Q3RE

Expert-verifiedFound in: Page 439

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

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

(e) Number of onto function =0 .

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

(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

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

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

(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 $={10}^{5}$ .

(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 $\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)\dots \dots ..(\mathrm{n}-...+1)$

If n=m

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

=n!

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

(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

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

Here,

m=5

n=10

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

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

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

(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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