Surjective sequence

I'm looking over countability of sets and if $x_n \subseteq X$ so that the sequence is a function from $\mathbb{N}$ to $X$ defined by $n \mapsto x_n, \forall n \in \mathbb{N}$

What is an example of a mapping to this sequence from the natural numbers that is not a surjection?

It looks like it has to be a surjection from the definition, as each number maps to some x_n. But the notes suggest otherwise.

$X$ is a set

X = {0, 1}, x_n = 0 for all n?

I also find it hard to believe what you've written is an accurate rendition of what your lecturer said.

Not quite sure what you're asking but couldn't you map all n to x_1 or n to x_2n? Does that answer your question?

I meant to write $(x_n)_{n=1}^{\infty}$ for the first bit.

What I mean is there is a sequence of elements of the set X which maps the natural numbers (starting from 1) to some x_n.

So there doesn't exist such a sequence for the set (0,1), say , but there does for the integers.

"A surjection from $\mathbb{N}$ to X corresponds to a sequence (x_n) $\subseteq X$ s.t. every element of X appears at least once in the sequence (X_n)"

So my question is, if such a sequence exists , how can it not be surjective (doesn't every element have to appear at least once) ?

Your clarifications are, I'm afraid, only making for more confusion. I'd encourage you to try and firm up your understanding of the definitions.



There are lots of sequences in {0,1} - i.e. lots of maps from N to {0,1}- uncountably many in fact. They are all surjective maps except

0,0,0,0,0,...
1,1,1,1,1,...



By the way a sequence in X isn't a subset of X - rather it's an element of $X^\infty$

I'm still not sure what "such" means here. Are you asking what sequences in X don't correspond to surjective maps from N to X? I've given two above in my previous comment.

I'm not saying you've said that, I'm saying it's (my best guess at) an example that isn't surjective.

Although it's so unclear what you actually mean that it quite possibly isn't a suitable example.

As RichE says, your "clarifications" are, if anything, only confusing the issue further.

From the definition, how can it not be surjective if n maps to x_n for all n?

Because {x_n : n \in N} need not cover the whole of X.

As I said before, define X = {0, 1}, x_n = 0 for all n, then {x_n : n \in N} = {0} which is a strict subset of X, therefore injective.

I note that you seem very unclear about the difference between X and {x_n : n \in N}.

Let X = {0,1}.

An example of a sequence in X is say

0,0,0,0,0,0,0,0,...

This corresponds to the function f:N -> X given by f(n) = 0 for all n or equally we might write x_n = 0 for all n.

The function f is not surjective as 1 is not in the image of f.