# Cardinality of the set of all functions from N to N

• Flying_Goat
In summary, the conversation discusses proving that the set of all functions from N to N has a cardinality of the continuum, and one approach involves using a function to map each function in the set to a number between 0 and 1. The conversation also raises questions about the injectivity of this mapping.
Flying_Goat

## Homework Statement

Let NN be the set of all functions from N to N. Prove that |NN|=c

## The Attempt at a Solution

I can prove that the set of all functions from N to {0,1} has cardinality of the continuum, but i can't generalise it. Any help would be appreciated.

Let f be a function from N to N. Construct a number, x, by writing a decimal point, then 0.f(1)f(2)f(3)... so that each function is mapped to the number, between 0 and 1, having the values of f as digits. That maps each function to a number. What numbers?

Thanks, I had thought of your argument when I was trying to prove |P(N)|=c but it didn't work out...I can't believe that it works for this question lol. Anyway thanks for your help.

Is the map injective? Because I could have f(2n-1)=12,f(2n)=3 or f(2n-1)=1,f(2n)=23 and both of these functions would give me 0.123123...

## What is the cardinality of the set of all functions from N to N?

The cardinality of the set of all functions from N to N, denoted as |NN|, is equal to the cardinality of the continuum, which is the cardinality of the real numbers. This means that there are uncountably infinite number of functions from N to N.

## Is the cardinality of the set of all functions from N to N the same as the cardinality of the set of real numbers?

Yes, the cardinality of the set of all functions from N to N is the same as the cardinality of the set of real numbers, both of which are uncountably infinite.

## Can the set of all functions from N to N be put into a one-to-one correspondence with the set of real numbers?

No, it is not possible to create a one-to-one correspondence between the set of all functions from N to N and the set of real numbers. This is because the cardinality of these two sets are both uncountably infinite, but they are not equal.

## Are there any functions from N to N that are not included in the set of all functions from N to N?

No, the set of all functions from N to N includes all possible functions from N to N. This is because the set of natural numbers is infinite, so there are infinite ways to map the numbers to itself.

## What is an example of a function from N to N?

An example of a function from N to N is f(n) = n+1, where the input is a natural number and the output is the next natural number. Another example is g(n) = n2, where the output is the square of the input.

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