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
Rina8888 [55]
3 years ago
14

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhib

it a one-to-one correspondence between the set of positive integers and that set.
a) the integers greater than 10
b) the odd negative integers
c) the integers with absolute value less than 1,000,000
d) the real numbers between 0 and 2
e) the set A×Z+ where A={2,3}
Mathematics
1 answer:
anzhelika [568]3 years ago
6 0

Answer:

a) Countably infinite

b) Countably infinite

c) Finite

d) Uncountable

e) Countably infinite

Step-by-step explanation:

a) Let S the set of integers grater than 10.

Consider the following correspondence:

f: S\rightarrow \mathbb{Z}^+ defined by f(10+k)=k-1 for k\in\mathbb{Z}^+/\{0\}.

Let's see that the function is one-to-one.

Suppose that f(10+k)=f(10+j) for k≠j. Then k-1=j-1. Thus k-j=1-1=0. Then k=j. This implies that 10+k=10+j. Then the correspondence is injective.

b) Let S the set of odd negative integers

Consider the following correspondence:

f: S\rightarrow \mathbb{Z}^+ defined by f(-(2k+1))=k.

Let's see that the function is one-to-one.

Suppose that f(-(2k+1))=f(-(2j+1)) for k≠j. By definition, k=j. This implies that the correspondence is injective.

c) The integers with absolute value less than 1,000,000 are in the intervals A=(-1.000.000, 0) B=[0, 1.000.000). Then there is 998.000 integers in A that satisfies the condition and 999.000 integers in B that satifies the condition.

d) The set of real number between 0 and 2 is the interval (0,2) and you can prove that the interval (0,2) is equipotent to the reals. Then the set is uncountable.

e) Let S the set A×Z+ where A={2,3}

Consider the following correspondence:

f: S\rightarrow \mathbb{Z}^+ defined by f(2,k)=2k, \;f(3,j)= 2j+1

Let's see that the function is one-to-one.

Consider three cases:

1. f(2,k)=f(2,j), then 2k=2j, thus k=j.

2. f(3,k)=f(3,j), then 2k+1=2j+1, then 2k=2j, thus k=j.

3.  f(2,k)=f(3,j), then 2k=2j+1. But this is impossible because 2k is an even number and 2j+1 is an odd number.

Then we conclude that the correspondence is one-to-one.

You might be interested in
Jerry cycles a 36-mile loop every day for two weeks. What is the total
Semmy [17]
The answer is: 18 miles.

: Hope this helps :)
5 0
3 years ago
Work out the surface area of this sphere.
MrRissso [65]

Answer:

452.2 cm

Step-by-step explanation:

A = 4πr²

A = 4 (3.14) (6)²

A = 4 (3.14) (36)

A = 452.16

A = 452.2 cm (nearest tenth)

5 0
2 years ago
A 26 foot ladder is set against the side of a house so that it reaches up 24 feet. If Mila grabs the ladder at its base and pull
olasank [31]

Answer:

13.9 ft

Step-by-step explanation:

4 0
2 years ago
Copland competes in 27 swimming events this season. He wins either first place or second place in each event. Copland has 3 more
marysya [2.9K]

Answer:

5

Step-by-step explanation:

3 0
3 years ago
Read 2 more answers
WILL AMKR BRAINLIEST IF GOTTEN RIGHTTT!!!!
Juliette [100K]

Answer:

1.06 sq inches

Step-by-step explanation:

its reasonable come on now

4 0
3 years ago
Other questions:
  • Determine which of the following terms is a function: a. (1,2) (2,4) (3,5) c. (1,2) (2,4) (2,6) b. (1,2) (2,3) (1,5) d. (1,2) (3
    7·1 answer
  • Caroline can sketch 3 cartoon strips in two hours. How long will it take her to sketch 6 strips?
    13·2 answers
  • Solve for the indicated variable t+m=r for t
    6·1 answer
  • What are the signs of the coordinates in Quadrant 1?​
    11·2 answers
  • Please help me with this​
    15·1 answer
  • IM TIMED HELP !!! GIVING 100 POINTS !! Triangles MNP and QST are congruent. What is the measure of angle T? all interior angles
    12·1 answer
  • The number line represents Allie’s scores in the first two rounds of a card game.
    5·1 answer
  • Mrs. Tart takes a survey in class. She discovers 60% of her students prefer using pens over pencils in class.
    15·2 answers
  • 5 3/4 divided by 2 1/4
    6·1 answer
  • ILL GIVE BRAINLIST ! Jamie spends the day at an amusement park. The table shows how many
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!