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Questions about set cardinalities (infinite/finite).

Mathematics Asked by Pwaol on January 13, 2021

I have a couple questions that I undertstood (maybe) intuitively and tried to prove them somehow, I would like if someone can give feedback and tell me if I’m doing it the right way.
True or false:
1)$ |{ frac {1} {2^n} mid nin mathbb{N} }| = |mathbb{N}|$
.
Intuitively this seems true, and I thought of proving it by introducing the function $f: mathbb{N} longrightarrow { frac {1} {2^n} mid nin mathbb{N} }$, $f(n)=frac {1} {2^n}$.

2) $K= {Ain P(mathbb{N}) mid A space is space finite } $, Assumption: $K$ is finite.

Intuitively I can see it’s infinite, since the power set of natural numbers have infinity elements, but I really don’t know how to prove it.

3) $L= {Ain P(mathbb{N}) mid A^c space is space finite }$,
Assumption $L$ is finite.
I think this is true, since for all $Ain P(mathbb{N})$, $A^c$ is infinite, so theres no elements in $L$.

I would really appreciate it if someone can approve my work, and tell me how to formalize the proofs, and point out my mistakes. Thanks in advance

One Answer

  1. Just give an infinite number of elements of $K$, for example all ${k}$ are in for $kin mathbb{N}$.

  2. If $A^c$ is the complementary of $A$, then I think the proposition is false, since the ${n; ngeqslant k}$ are in $L$ for all $k$.

Correct answer by math on January 13, 2021

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