Articles of elementary set theory

Indicator Function Distributive Property Proof

This is my first post(: I’m trying to understand how to prove the distributive property using the indicator function. I have made the truth tables and understand how this is proved using set notation as in this question: Set Distributive Property Proof But I cant seem to understand how to write this using indicator function […]

A nonempty class of isomorphic groups defines a group

The context of this question is from the definition of the sporadic Mathieu group $M_{23}$, which (in one possible definition) is the stabilizer of a point in $M_{24}$, which is a certain subgroup of $S_{24}$ (a permutation group on $24$ points). When I read this I was a bit surprised since they haven’t specified which […]

Give a bijection $f: (c,d) \to \Bbb R$ (f no trigonometric) to prove every open interval has the same cardinality of R

This question already has an answer here: Prove: Any open interval has the same cardinality of $\Bbb R$ (without using trigonometric functions) 5 answers

Proof of A $\cap$ $B^c$ = $\emptyset$ $\implies$ A $\subseteq$ B

Is this reasoning correct? I have to prove A $\cap$ $B^c$ = $\emptyset$ $\implies$ A $\subseteq$ B By taking the contrapositive, we get A $\not\subseteq$ B $\to$ A $\cap$ $B^c$ $\not=$ $\emptyset$ $\iff$ A $\not\subseteq$ B $\to$ (A $\cap$ $B^c$ $\not\subseteq$ $\emptyset$) $\lor$ ($\emptyset$ $\not\subseteq$ A $\cap$ $B^c$) We can discard $\emptyset$ $\not\subseteq$ A $\cap$ […]

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 […]