Intereting Posts

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$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$…
If $\alpha=\frac{2\pi}{7}$,prove that $\sin\alpha+\sin2\alpha+\sin4\alpha=\frac{\sqrt7}{2}$
how do you find the modular inverse
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units of group ring $\mathbb{Q}(G)$ when $G$ is infinite and cyclic
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It is known that closed sets of $\mathbb{R}$ satisfies continuum hypothesis, that is, every closed subset of $\mathbb{R}$ is either countable or of the cardinality of

the continuum.

Is the cardinality of uncountable $G_{\delta}$ set of $\mathbb{R}$ equals the cardinality of the continuum?

- axiom of choice: cardinality of general disjoint union
- The cardinality of the set of countably infinite subsets of an infinite set
- Are there any infinites not from a powerset of the natural numbers?
- What is the cardinality of a set of all monotonic functions on a segment $$?
- Division of cardinals
- How to prove cardinality equality ($\mathfrak c^\mathfrak c=2^\mathfrak c$)

- Not a small, not a big set
- Proof of cardinality inequality: $m_1\le m_2$, $k_1\le k_2$ implies $k_1m_1\le k_2m_2$
- Do sets whose power sets have the same cardinality, have the same cardinality?
- What is the cardinality of the set of all sequences in $\mathbb{R}$ converging to a real number?
- There's non-Aleph transfinite cardinals without the axiom of choice?
- Subsets of the reals when the Continuum Hypothesis is assumed false
- For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
- Showing that $|\mathcal{P}(\mathbb{N})| = |\operatorname{Maps}(\mathbb{N},\mathbb{N})|$
- What's application of Bernstein Set?
- The cardinality of the set of countably infinite subsets of an infinite set

Yes. In fact the cardinality of every Borel set is countable or size continuum.

There is an easy proof for $G_\delta$. Every $G_\delta$ set is completely metrizable, and therefore either countable or contains a perfect set (which has a copy of a Cantor space).

(See this answer for the former fact, and the Cantor-Bendixson derivative, and related theorems for the latter)

- G acts on X transitively, then there exists some element that does not have any fixed points
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- Do we have always $f(A \cap B) = f(A) \cap f(B)$?
- How to solve $100x +19 =0 \pmod{23}$
- Proving Functions are Surjective
- Counting number of solutions with restrictions
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- Mean and Variance of Methods of Moment Estimate and Maximum Likelihood Estimate of Uniform Distribution.
- the converse of Pythagoras Theorem
- A problem about periodic functions
- Proof by induction: $\sum\limits_{i=1}^{n} \frac{1}{n+i} = \sum\limits_{i=1}^{n} \left(\frac{1}{2i-1} – \frac{1}{2i}\right)$
- Splitting of prime ideals in algebraic extensions
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