Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

a) Define what it means for a function from the set of positive integers
to the set of positive integers to be one-to-one
b) Define what it means for a function from the set of positive integers to the set
of positive integers to be onto.
c) Give an example of a function from the set of positive integers to the set of
positive integers that is both one-to-one and onto.
d) Give an example of a function from the set of positive integers to the set of
positive integers that is one-to-one but not onto.
e) Give an example of a function from the set of positive integers to the set of
positive integers that is not one-to-one but is onto.
f) Give an example of a function from the set of positive integers to the set of
positive integers that is neither one-to-one nor onto.

a) Function f is one-one if $f (a) = f (b)$ implies that $a = b$ for all and in the domain.

b) Function $f : A \to B$ is onto if for every element there exist an element $a \in A$ such that $f (a) = b$

c) Required answer is $f (n) = n$ .

d) Required answer is $f (n) = 2 n$

e) Required answer is $f (n) = \{\begin{cases} (\frac{n}{2}) if n even \\ (\frac{n + 1}{2}) if n odd \end{cases}$

f) Required answer is $f (n) = 1$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Definition of funtion

A function $f : A \to B$ has the property that each element of a has been mapped to exactly one element of B.

Step 2: One-One function

a) The function $f : A \to B$ is one to one if and only if $f (a) = f (b)$ implies that a=b for all a and b in the domain.

Step 3: Onto function

b) The function $f : A \to B$ is onto if and only if for every element $b \in B$ there exist an element $a \in A$ such that $f (a) = b$

Also, function f is onto if range of function is equal to co-domain of function.

Step 4: One-One and onto function

c) We need to find a function from the positive integers to the positive integers $f : Z^{+} \to Z^{+}$

such that function is both one –one and onto.

For example,

$f (n) = n$

is one to one let $f (a) = f (b) \Rightarrow a = b$

Is onto since for all $b \in Z^{+}$ , there exists $a \in Z^{+} : f (a) = b$

Step 5: One-One and but not onto function

(d) We need to find a function f from the set of positive integers to the set of positive integers $f : Z^{+} \to Z^{+}$

such that function is one –one but not onto.

For example,

$f (n) = 2 n$

f is one to one let $f (a) = f (b)$ ,by definition of $f : 2 a = 2 b$ . Dividing each side of the equation by 2, we get a=b

is not onto: let $b \in z^{+}$ , then we $a = \frac{b}{2}$ is the only value with the property $f (a) = f (\frac{b}{2}) = 2 (\frac{b}{2}) = b$

However, if b is odd, then $a = \frac{b}{2}$ is not a positive integer $(a \notin Z^{+})$ and thus f is not onto.

Step 6: Not one-one but onto function

We need to find a function f from the set of positive integers to the set of positive integers

$f : Z^{+} \to Z^{+}$

such that function is not one –one but onto.

For example,

$f (n) = \{\begin{cases} (\frac{n}{2}) if n even \\ (\frac{n + 1}{2}) if n odd \end{cases}$

Thus, the image of and is then, the image of and is , the image of and is , the image of and is , and so on.

is not one to one because n=1,2 have the same image.

is onto since range is equal to co-domain.

Step 7: Neither one-One and onto function

We need to find a function f from the set of positive integers to the set of positive integers such that

$f : Z^{+} \to Z^{+}$

And f is neither one-one nor onto.

For example, let

$f (n) = 1$

is not one to one because all positive integers have the same image.

is not onto because all positive integers (except for 1) are not in the range.

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Discrete Mathematics and its Applications

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Short Answer

Step by Step Solution

Step 1:Definition of funtion

Step 2: One-One function

Step 3: Onto function

Step 4: One-One and onto function

Step 5: One-One and but not onto function

Step 6: Not one-one but onto function

Step 7: Neither one-One and onto function

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Step 6: Not one-one but onto function

Step 7: Neither one-One and onto function

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