Articles of elementary set theory

Cardinality of infinite family of finite sets

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 […]

Help with an elementary proof regarding cardinalities of finite sets.

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 […]

Set theory, property of addition of natural numbers in the cardinal way

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 = […]

Problem understanding “and”,“or” and importance of “()” in set theory

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 […]

Showing the consistency of an equivalence relation over *

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 […]

Proof of $X \times X \hookrightarrow X$ implies $^2 \hookrightarrow X$

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 […]

Are $\prod_{i\in I}{X_i^2}$ and $(\prod_{i\in I}{X_i})^2$ the same?

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: $$\prod_{i\in I}{X_i^2}=\left(\prod_{i\in I}{X_i}\right)^2$$

Regular cardinals and unions

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 […]

How do I understand the null set

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?

Cantor's theorem via non-injectivity.

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 $F$ […]