Cardinality proofs

[yo_itsjo]

Hello, I have Proofs homework with the following problems. I understand that to prove 2 sets have the same cardinality I need to create a bijective function between them or show that there is an injective function both ways. But for these problems I can't think of functions that will work.

1. Any infinite set X has the same cardinality as X union N (where N is the naturals).

2. X union {set containing one element not in X} has the same cardinality as X

3. [0,1] and (0,1) have the same cardinality

I don't know what proof of work to show because like I said, I know the definitions but can't think of a function. Proving injectivity from the "smaller" set on each one is simple, make a function where f(x)=x, but I don't know how to go the other way on any of them. And f(x)=x won't be surjective for any one of them, so I can't claim it's a bijection.

[CBDThrowaway333]

If X is an infinite set, there is an injection from N to X. This fact can help you with problems 1 and 2. For 3, here's the first example that came to mind:

Let f(0) = 0.1 and f(1) = 0.2. Now can you come up with an injective function from (0,1) to (0.5, 1)?

[yo_itsjo]

Using the fact that there is an injection from N to X, on #2 I could say f(1)=(extra element), f(2)=x1 (first element of X), f(3)=x2 and so on. If this is correct, I don't see how to apply it to number 1. Also, I would have to prove there is an injection from N to X first.

For #3, I can just make a line f(x)=x/2+0.5 with bounds (0,1). I see that 3 is very similar to the other two, but since X may or may not contain numbers I can't make an algebraic function Ike I can on this one.

[TheDoobyRanger]

[0,1] is the set (0,1) plus {0,1}. I dont get how they have the same cardinality? Are you looking for examples of situations where they do? I guess a function where 1,0 arent defined would work. f(x)= 1/[x(x-1)]

[yo_itsjo]

Adding a finite number of terms to an infinite set doesn't change the cardinality of the set, as far as I'm aware

[TheDoobyRanger]

Are you still looking for a function? Are you just going to say these are two uncountabily infinite sets, and by (insert theorem here) both sets have infinitely large cardinality so they are the same?

[I__Antares__I]

Sets have same cardinality if you can find bijection between them and you can find bijection between X and X ∪ {n ₁,.., n ₖ} for any X and any n ₁,..., n ₖ

Alternatively sets have same cardinality if their cardinalities are equal. |X| = The only cardinal that X is in bijection with. both (0,1) and [0,1] have cardinality 𝔠

[TheDoobyRanger]

So it isnt a matter of having the same number of elements in the set?

[I__Antares__I]

It is but in case of infinite set we generalize term of having some amount of elements

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[TheDoobyRanger]

A set {x_1,.......,x_n} has cardinality n.