Inspiration for the following question comes from an exercise in Spivak’s Calculus, there too are considered finite sets of real numbers in interval $[0,1]$ but in completely different setting. I will state formulation of the question and my attempt to solve it. I should note that all my knowledge of set theory mostly comes from […]

I was relatively confused trying to produce a proof of this theorem, however, I have provided my attempt. I would greatly appreciate it if people can help steer me to the correct proof, or provide a simple fix to my proof if it is near correct. Proposition: If $X$ is a finite set of cardinality […]

Consider the set of natural numbers $\mathbb N$. On this set we define an operation ‘+’, as follows: for all $n,m \in \mathbb N$ we put $n+m$ to be the unique natural number $t \in \mathbb N$ such that there are $X,Y$ such that $$ X \cap Y = \emptyset, \quad X \cup Y = […]

I was reading the distributive law of sets (I keep coming back to basic maths when needed, forget it after some time, then come back again. Like I’m in loop): $A\cup(B \cap C)=(A\cup B)\cap(A\cup C)$ The proof (which I’m assuming everyone knows) has a transaction between lines which baffled me , which are: $x \in […]

Let $E$ be an equivalence relation on the set of all ordered pairs of non-negative integers ($N\times N$). It is defined as $$(a,b)E(x,y) \Longleftrightarrow a+y = b+x$$ Multiplication ($*$) is defined as $$(a,b)*(x,y) = (ax+by, ay+bx)$$ Without using substraction or division, how can I show that $E$ is consistent with $*$ ? By consistent I […]

Consider the following two statements (where $[X]^2$ denotes the set of all unordered two-element subsets of $X$): (HSO) For every infinite set $X$ there exists an injection $f: X \times X \hookrightarrow X$ (HSU) For every infinite set $X$ there exists an injection $f: [X]^2 \hookrightarrow X$ The following is an exercise from a book […]

We have: $$\prod_{i\in I}{X_i}=\left\{f:I\to\bigcup_{i\in I}{X_i}~\Big|~ (\forall i\in I)\big(f(i)\in X_i\big)\right\}$$ Is it true: $$\left|\prod_{i\in I}{X_i^2}\right|=\left|\left(\prod_{i\in I}{X_i}\right)^2\right|$$ and can we assume: $$\prod_{i\in I}{X_i^2}=\left(\prod_{i\in I}{X_i}\right)^2$$

If a cardinal $\kappa$ is regular then it cannot be written as a union of fewer than $\kappa$ sets, each of size less than $\kappa$. This seems to be a very useful characterization. I have seen a proof or two, but can’t grasp all the details… I am horrible with ordinal and cardinal arithmetic. Could […]

from my understanding,every set has at least two subsets; the null set and the original set itself. My question is, what is the power set of the null set? Shouldn’t it be just itself?

The usual proof of Cantor’s theorem proceeds as follows. Let $X$ denote a set and consider a function $F : X \rightarrow \mathcal{P}(X)$. Then we define $D \in \mathcal{P}(X)$ by writing $D = \{x \in X \mid x \notin F(x)\}$, observe that $D$ is not in the image of $F$, and thereby conclude that $F$ […]

Intereting Posts

For an outer measure $m^*$, does $m^*(E\cup A)+m^*(E\cap A) = m^*(E)+m^*(A)$ always hold?
Evaluating a triple integral
Suppose that $(s_n)$ converges to s. Prove that $(s_n^2)$ converges to $s^2$
Is the set of all functions from $\mathbb{N}$ to $\{0,1\}$ countable or uncountable?
The field of Laurent series over $\mathbb{C}$ is quasi-finite
Rational points on $y^2 = 12x^3 – 3$.
Is a completion of an algebraically closed field with respect to a norm also algebraically closed?
How to calculate $|f|_{0}$?
Abstract algebra book recommendations for beginners.
Why do the $n \times n$ non-singular matrices form an “open” set?
Normal approximation of tail probability in binomial distribution
Is it possible to make the set $M:=\{(x,y)\in\mathbb{R}^2:y=|x|\}$ into a differentiable manifold?
Differential of transposed matrices
Some three consecutive numbers sum to at least $32$
Find the largest prime factor