Question

Suppose A = {3n | n E Z} and f: R --> R is defined by f(x) = X^(x). f is the characteristic function of set A.

(a) What is f ({2n | n E Z})?

(b) What is f^-1 ({2n | n E Z})?

Answer 1

i am kind of lost
A are integers divisible by 3, and \(f(x)=x^x\) right?

Answer 2

see the screenshot. it is a confusing question

Answer 3

oooh i see

Answer 4

The characteristic function of the set A, \(\chi_A(x)\) is defined as: $$\chi_A(x)=1\quad\text{if } x\in A$$and: $$\chi_A(x)=0\quad\text{if } x\not{\in} A$$

Answer 5

characteristic function means \(f(n)=1\) if \(n\in A\) and \(f(n)=0\) if \(n\notin A\)

Answer 6

what @nerdguy2535 said

Answer 7

The question is asking what is the image and pre-image of the set $$\{2n\mid n\in\mathbb{Z}\}$$ under the characteristic function.

Answer 8

So for a), they want to know what the image of \(\{2n\mid n\in \mathbb{Z}\}\) is under f. In other words, if \(x\in \{2n\mid n\in \mathbb{Z}\}\), what are the possible answers for \(f(x)\)?

Answer 9

you got this?

Answer 10

umm not yet...

Answer 11

image of 2n would be {2, 4, 6,...} and preimage would be 2?

Answer 12

i think the image is 1 and 0, since it is the characteristic function on integers divisible by 3

Answer 13

1 and 0?

Answer 14

the numbers that would get sent to 1 in 2n are the ones that are divisible by 6

Answer 15

since they have to be divisible by 3 to get sent to 1, and all are even

Answer 16

I am not sure if I understand it. can we go back to the beginning?

Answer 17

sure
the notation used is a little weird

Answer 18

Suppose A = {3n | n E Z} and f: R --> R is defined by f(x) = X^(x). f is the characteristic function of set A.

(a) What is f ({2n | n E Z})? // Do I need to find preimage and image of this?

Answer 19

the set A is \(\{...-6,-3,0,3,6,9,...\}\)

Answer 20

the function sends those numbers to 1 so \(f(-6)=1,f(-3)=1,f(0)=1,f(3)=1\) etc

Answer 21

it sends everything else to \(0\) so \(f(1)=0,f(2)=0.f(\pi)=0\) etc

Answer 22

I am sorry I am a little lost. where did f(-6) = 1 comes from?

Answer 23

\[A=\{3n|n\in \mathbb{Z}\}\] so if \(n=-2\) then \(3n=-6\)

Answer 24

A is the set of integers divisible by 3

Answer 25

and since \(-6\) is divisible by 3, it is in A and so \(f(-6)=1\)

Answer 26

ok what does it have to do with this "What is f ({2n | n E Z})?"

Answer 27

the way i interpret the question, is what does \(f\) do to even numbers ?

Answer 28

square it?

Answer 29

f: R --> R is defined by f(x) = X^(x) <-?

Answer 30

but then maybe i don't really understand the question
i would say it sends numbers divisible by 6 to 1, all others to zero

Answer 31

oh ok. I am sorry I still don't understand but thanks for trying to help

Answer 32

yw, sorry i could not do better