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
If a university wants to maintain a 14:1 ratio between students and teachers, how many teachers would be needed to accommodate 8
zheka24 [161]
<span>So we want to know the number of teachers the university should have if the ratio of students to teachers is 14:1 and the number of students is 896. If x is the number of teachers: 14/1=896/x. Now we solve for x: Lets multiply both sides with x: x*(14/1)=896 and divide both sides by 14 and we get: x=896/14 and x=64. So the university should have 64 teachers.</span>
7 0
3 years ago
Read 2 more answers
The expression represents the cost to purchase tickets for a play, where is the number of tickets. Be prepared to explain your r
sp2606 [1]

Answer:B

Step-by-step explanation: It is wrong bruv

4 0
2 years ago
Jackie follow the steps to elevate the expression 15 - (37 + 8) divide 3 37 + 8 = 45 45-15= 30 30 divided by 3 = 10 Mark looks a
damaskus [11]

Answer:

So, we conclude that Mark is right.

Step-by-step explanation:

If we have the following expression: 15 - (37 + 8) : 3.

We will first divide the value we get in parentheses and then subtract it.

Because sharing is an older mathematical operation than subtraction. Therefore, we get:

15-(37+8):3=15-45:3=15-15=0

So, we conclude that Mark is right.

3 0
3 years ago
Calculate the radius<br>22cm​
sweet [91]

Answer:

11

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Evaluate the expression.<br> -6+3(-3+7).
lesantik [10]

Answer:

answer for this question is 6

3 0
2 years ago
Read 2 more answers
Other questions:
  • There are 37 student that are riding in cars for a field trip. If a car can carry 4 students, how many cars are necessary to tak
    7·1 answer
  • 5d^9\4(d)^2. Simplify
    12·1 answer
  • Please help me thank you.
    13·1 answer
  • Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the function: y=4x^2-8x-3
    13·1 answer
  • Embarcación averiada. Dos patrullas de la Guardia Costera de Guatemala están separadas 225 millas y viajan una en dirección de l
    8·1 answer
  • 1/2(b-3)=5/2 solve equation plzzz
    6·1 answer
  • Peter buys a small wind turbine for use on his farm. It generates 13,000 kWh of electricity in the first year. He uses 7,570 kWh
    13·1 answer
  • -<br>(23)<br>x +. 8<br>1 - 2x = 8 + x<br>5<br>3<br>-<br>5x - 1<br>24​
    15·1 answer
  • A chef cooked 4 kilograms of mashed potatoes for a dinner party. If the guests only ate
    6·2 answers
  • Find the midpoint of the segment below and enter its coordinates as an ordered pair. If necessary, express the coordinates as fr
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!