Mathematics, 10.07.2019 04:00 live4dramaoy0yf9

Prove that the set of even integers is denumerable

Answers: 1

Show answers

Answers

to show that the set of all even integers is denumerable, we prove that there is a bijection from $\mathbb{z}^+$ to $2\mathbb{z}$ , where $2\mathbb{z} = \{2k \mid k \in \mathbb{z}\}$ , the set of all even integers.

proof. note that the set of even integers can be put into a list, that is, $2\mathbb{z} = \{0,2,-2,4,-4,6,-6,\ldots\}$ . we can intuitively see that all the even integers can be listed, allowing for a way to count all the even integers (however, we will never finish counting). we can be more rigorous in our argument by finding an explicit bijection. we can think about how to recreate the list we have made.

let the function $f: \mathbb{z}^+\to 2\mathbb{z}$ be defined by

$f(x) = \begin{cases}x & \text{ if $x$ is even} \\-(x-1) & \text{ if $x$ is odd.}\end{cases}$

for example, $f(1) = -(1-1) = 0$ , $f(2) = 2$ , $f(3) = -(3-1) = -2$ , $f(4) = 4$ , and so on. clearly this function generates this list. now we show that $f$ is a bijection. to do this, we must show that $f$ is injective and surjective.

injectivity

to show that $f$ is injective, we must show that for any $a,b\in\mathbb{z}^+$ , $f(a) = f(b)$ implies $a=b$ .

suppose that $a,b\in\mathbb{z}^+$ such that $f(a) = f(b)$ . then we have four cases:

case 1: if $a,b$ are both odd, then

$f(a) = f(b) \implies -(a-1) = -(b-1) \implies a - 1 =b - 1\implies a = b.$

case 2: if $a,b$ are both even, then

$f(a) = f(b) \implies a=b.$

case 3: if $a$ is even and $b$ is odd, then

$f(a) = f(b) \implies a=-(b-1) \implies -a = b-1.$ .

but since $a$ is a positive even integer, we have $a \ge 2 \implies -a \le -2$ and since $b$ is a positive odd integer, we have $b \ge 1 \implies b - 1 \ge 0$ .

we inferred that $-a = b-1$ is true in this case, but the left- and right-hand inequalities actually do not agree. this is a contradiction, so case 3 cannot happen.

case 4: if $a$ is odd and $b$ is even, then by similar argument as in case 3, case 4 cannot happen.

hence $f$ is injective.

surjectivity

now we show that $f$ is a surjection. to do this, we must show that for any $b \in 2\mathbb{z}$ , there is $a \in \mathbb{z}^+$ such that $f(a) = b$ . we must assert that there is an element in the domain that has $b$ as its image. so suppose we have $b \in 2\mathbb{z}$ . here, $b$ is even.

case 1: if $b > 0$ , then $b\in\mathbb{z}^+$ and $f(b) = b$ , since $b$ is even.

case 2: if $b < 0$ , then $-b > 0 \implies -b + 1 > 1$ so $-b + 1\in\mathbb{z}^+$ and $f(-b + 1) = + 1 - 1) = ) = b$ , since $-b + 1$ is odd.

case 3: if $b = 0$ then $f(1) = -(1-1) = 0$ .

so $f$ is also a surjection.

therefore, $f: \mathbb{z}^+\to 2\mathbb{z}$ is a bijection because it is injective and surjective.

it follows that $2\mathbb{z}$ , the set of all even integers, is denumerable.

Answer from: Quest

x^6

step-by-step explanation:

this is correct because:

x-6=0

0*x=0

x^6

Answer from: Quest

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 22:50

Aweekly newspaper charges $72 for each inch in length and one column width. what is the cost of an ad that runs 2 column inches long and 2 columns wide? its not 288

Answers: 1

Answer

Mathematics, 22.06.2019 00:50

There are 8 movies that you would like to see currently showing in theatres. in how many different ways can you choose a movie to see this saturday and one to see this sunday?

Answers: 1

Answer

Mathematics, 22.06.2019 01:30

Sara is flying a kite at the park. the angle of elevation from sara to kite in the sky is 53° and the length of the string is 32 feet. of sara is 5 feet tall how far off the ground is the kite

Answers: 1

Answer

Mathematics, 22.06.2019 02:00

An amusement park manager determined that about 23 of all customers would wait in long lines to ride the new roller coaster. which simulation could not be used to answer questions about whether a customer would wait?

Answers: 1

Answer

You know the right answer?

Prove that the set of even integers is denumerable...

Questions

Mathematics, 10.11.2020 02:20

The manager of a campus bookstore finds that in the last hour before closing, 41 students bought two or more textbooks. Of these 4 used cash only, 5 u...

Mathematics, 10.11.2020 02:20

Mr. Smith is adding food coloring to water to create his own paint colors. He has 5 1/3 cups of water. to create a color, he needs 2/3 cups of water....