Articles of elementary set theory

common knowledge and concept of coarsening partition

Here is a proof of the equivalence between my definition and Aumann’s for “common knowledge”. I’m assuming some familiarity with set partitions. Aumann’s definition is in terms of the Kolmogorov model of probability. In particular, a proposition is identified with the set of possible worlds in which the proposition is true. Let P₁ be that […]

Ordinal Fractions

Is any fraction $\,{x}\big/ {y}\,$ an ordinal number and if so, does ordinal $\,1 = \big\lbrace0,\dots,y – 1\big/y\big\rbrace\,$ instead of $\,\left\lbrace0\right\rbrace\,$? “If (X, <=) is a well ordered set with ordinal number x, then the set of all ordinals < x is order isomorphic to X. This provides the motivation to define an ordinal as […]

Does $f\circ g$ injective imply $f$ injective for functions $f,g:A\to A$?

I have some problem with proof of the following question: I have two functions $f,g: A \to A$, I know that $(f \circ g)$ is injective, is it possible at all to prove that $f$ is injective? I know that if $g: A \to B$ and $f: B \to C$, I can find example which […]

Understanding $A^A$ in set theory

If $A=\left\{1,2,3\right\}$ so what will be $A^A$? What is the geometric interpretation for $A^A$? Thanks

Possible divisors of $s(2s+1)$

I write $\psi(s) = s(2s+1)$ and let $d$ be the divisor function. If $s$ is prime then 4 divides $d(\psi(s))$. For example if $s=37$ then $d(\psi(s)) = d(2775) = 12$ and $4|12$. Is this trivial? I am not sure how to attack and prove this. Here is my approach: Now I am thinking $d$ is […]

Cardinality of a $\mathbb Q$ basis for $\mathbb C$, assuming the continuum hypothesis

Prove that $\operatorname{tr.deg}(\mathbb{C/Q})=\mathfrak{c}$ (where $\mathfrak{c}$ is the cardinality of $\mathbb{R}$) using the continuum hypothesis. $Proposition$ If $E/F$ is an algebraic extension then $|E|=\aleph_0 |F|$ Proof: Let $S$ be a trancendental basis of the extension $\mathbb{C/Q}$.Then $|S| \leqslant \mathfrak{c}$.Suppose that $|S|< \mathfrak{c}$. Then, by the continuum hypothesis, $S$ is finite or countable. Let $S=\{s_1,s_2, \ldots \}$ […]

Is $\mathbb R^2$ equipotent to $\mathbb R$?

This question already has an answer here: Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ 2 answers

What is Needed for Cantor-Bernstein-Schröder Theorem

The $CBS$ Theorem is a superb tool to prove the equality of size of two infinite sets. Inject $A$ in $B$, inject $B$ in $A$, done. When thinking in naïve set theory one often is unaware of the full extent of one’s assumptions. What are the axioms (fragment of $ZFC$) required to allow for the […]

how to show that a subset of a domain is not in the range

If $g$ is a function from a set $Y$ to the collection of all subsets of $Y$, how does one show that there exists a subset of $Y$ that is not in the range of $g$? An explanation of how to attempt a proof would be great. Thanks.

How many reflexive binary relations there are on a finite countable set?

We know that binary relation is subset of Cartesian product made by set on to itself. let’s say we have a set with two elements $A=\{0,1\}$ So Cartesian product is $C=A\times A = \{(0,0),(0,1),(1,0),(1,1)\}$ Then we should write down all subsets of this product to get all binary relations. Right? $1. \ \ \ \ […]