1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
iris [78.8K]
3 years ago
15

20. Give an example of a function from N to N that is a) one-to-one but not onto. b) onto but not one-to-one. c) both onto and o

ne-to-one (but different from the identity function). d) neither one-to-one nor onto.
Mathematics
1 answer:
Oduvanchick [21]3 years ago
8 0

Answer:

Step-by-step explanation:

a) To provide an example of a function N → N that is one-to-one but not onto.

Suppose f:N\to N  to be f(n)=n^2

Then; \text{a function } \ f: A \to B\  \text{is one-to-one if and only if } f(a) = f(b) \implies a = b \ for \ a, b  \ \epsilon \ A.

\text{a function } \ f: A \to B\  \text{is onto if and only if  for every element } b  \ \epsilon \ B  \\ \text{there exist an element a}  \ \epsilon\  A \ such \  that f(a) = b}

Now, assuming a \ \Big {\varepsilon}  \ N \&  \ b  \ \epsilon  \ N;

Then f(a) = f(b)

a^2 =  b^2 \\ \\ a = b

The above function is said to be one-to-one

\text{it is equally understandable that not every natural number is the square of a natural number}e.g

2 is not a perfect square, hence, it is not regarded as the image of any natural no.

As such, f is not onto.

We can thereby conclude that the function  f(n) = n^2 is one-to-one but not onto

b)

Suppose f: N \to N be

f(n) = [n/2] \\ \\  For \ n =1, f(1) = [1/2] = [0.5] = 1 \\ \\ For \ n=2 , f(2) = [2/2] = [1] = 1

It implies that the function is not one-to-one since there exist different natural no. having the same image.

So, for n \epsilon N , there exists an image of 2n in N

i.e.

f(2n) = [2n/2] = [n] = n

Hence, the function is onto

We thereby conclude that the function f(n) = [n/2] \text{ is onto but not one-to-one}

c)

let f: N\to N be  f(n) = \left \{ {{n+1, \ if \ n \ is \ even } \atop n-1 , \ if \ n \ is \ odd} \right.

So, if n, m is odd:

Then:

f(n) = f(m) \\ \\ n-1 = m-1 \\ \\ n = m

Likewise, if n, m is even:

Then;

f(n) = f(m) \\ \\ n+1 = m+ 1  \\ \\ n = m

The function is then said to be one-to-one.

However, For n \epsilon N and is odd, there exists an image of n - 1that is even;

f(n - 1) = n -1 + 1 =n

For n \epsilon N and is even, there exists an image of n + 1that is odd;

f(n - 1) = n +1 - 1 = n

where(; implies such that)

Hence, this function is said to be onto.

We can therefore conclude that the function f(n) = \left \{ {{n+1, \ if \ n \ is \ even } \atop n-1 , \ if \ n \ is \ odd} \right. is both onto and one-to-one.

d)

Here, to provide an example where the f:N \to N is neither one-to-one nor onto.

SO;

Let f : N \to N is defined to be f(n)=0

Then, since every integer has the same image as zero(0), the function is not one-to-one.

Similarly, the function is not onto since every positive integer is not an image of any natural number.

We, therefore conclude that, the function f(n)=0 is neither one-to-one nor onto.

You might be interested in
A rental car company charges $42.72 per day to rent a car and $0.10 for every mile driven. Zoey wants to rent a car, knowing tha
Colt1911 [192]

Answer:

7 Days

Step-by-step explanation:

Miles cost:

$0.10 * 200 = $20


Daily Cost

Her budget at this point is $330 because $350 - $20 (the miles)

$330 / 42.72  = Roughly 7.72

Therefore Zoey can drive for 7 days before her budget runs out.

6 0
3 years ago
Using the protractor tool on the right, choose the image that represents an angle measuring 55°.
Anarel [89]

Answer:

I think it's C.

Step-by-step explanation:

hope you get it right

5 0
1 year ago
lance bought a square postage stamp to a mail a card to his cousin. if the stamp has 4 square centimeters.what was the dimension
Lubov Fominskaja [6]

The dimension of one side of the stamp was 2 centimeters

Step-by-step explanation:

The formula of the area of any square is A = l², where l is the

length of the side of the square

lance bought a square postage stamp to a mail a card to his cousin

∵ The stamp has 4 square centimeters

∴ The area of the stamp = 4 cm²

∵ The stamp shaped a square

∵ The formula of the area of the square is A = l²

- Equate the formula of the area of the square by the area of the stamp

∴ l² = 4

- Take √ for both sides to find l

∴ l=\sqrt{4}

∴ l = 2 cm

The dimension of one side of the stamp was 2 centimeters

Learn more:

You can learn more about the area of the shapes in brainly.com/question/10677255

#LearnwithBrainly

5 0
3 years ago
The first four terms of a sequence are shown below.
devlian [24]

Answer:

820.125

Step-by-step explanation:

2*4+2/2=<em>9</em>

9*4+9/2=<em>40.5</em>

40.5*4+40.5/2=<em>182.25</em>

182.25 * 4 + 182.25/2 = 820.125

8 0
3 years ago
I have been asking and asking for help and not one hleped me all they did was troll and say dumb stuff that has nothing to do wi
Andreas93 [3]

Answer:

Try 120

Step-by-step explanation:

There are five dashes in between 100 and 200. Count up by 20s

6 0
3 years ago
Other questions:
  • Simplify the expression.<br><br> (−0.75x + 8) + (1.5x + 4.4)<br><br> The simplified expression is
    7·1 answer
  • the larger triangle is a dilation of the smaller triangle with a center of dilation at (2,-1). Whatbis the scale factor of the d
    5·1 answer
  • I need help with this I have no idea what graph goes with the equation?
    7·1 answer
  • Round 0.1347 to the nearest tenth. Do not write extra zeros.
    15·2 answers
  • Well he buys two books at a bookstore. He spent $44.52 after 6% sales tax is applied, but the clerk does not provide him with th
    5·1 answer
  • Can someone know this asap​
    14·1 answer
  • What is the y intercept of the line y = 2x+1?
    7·2 answers
  • A local sawmill sells a total of 250,000 board feet of hardwood each year. The ratio of softwood sold to hardwood sold is 5: 3.
    13·1 answer
  • If m varies directly as n and m = -4 when n = 32, find m when n = 3
    15·1 answer
  • <img src="https://tex.z-dn.net/?f=1%20-%2025x%20%7B%7D%5E%7B2%7D%20" id="TexFormula1" title="1 - 25x {}^{2} " alt="1 - 25x {}^{2
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!