Articles of elementary set theory

Order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$

How can one prove the existence of an order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$? Can you give an example of such a bijection?

Direct formula for elements of power set?

Let $A_n$ be an ordered set: $$A_n = \{ 1,2,3,\dots,n\}$$ Then the powerset of $A_n$ lets call it $P_n$, is $$P_n=\{\emptyset,\{1\},\{2\},\dots,\{1,2\},\{1,3\},\dots,\{1,2,3,\dots,n\}\}$$ How can I find the formula $F(i,j)\colon\mathbb{N}\times\mathbb{N} \to \mathbb{N}$, that given $i$ and $j$ will return $P_n[i][j]$, that is $j$’th element in $i$’th set of $P_n$? I searched google but couldn’t find anything like […]

Clarify definitions of relation and 0-ary relation

From mathworld.wolfram.com: A relation is any subset of a Cartesian product But if so, then the null set is all of: 0-ary, 1-ary, 2-ary etc. Wouldn’t it be better to define it as: A relation is any non-empty subset of a Cartesian product Second question, from this answer I understand that in set theory a […]

On proving $\{0,1\}^\mathbb{N}\sim\{0,1,2\}^\mathbb{N}$

I’m trying to prove $\{0,1\}^\mathbb{N}\sim\{0,1,2\}^\mathbb{N}$ (the sets are equinumerous). I have already proved that $\{0,1\}^\mathbb{N}\sim\mathbb{N}^\mathbb{N}$, with the following method: For every countable $X\subset\{0,1\}^\mathbb{N}$, $\{0,1\}^\mathbb{N}\sim\{0,1\}^\mathbb{N}\setminus X$. There exists an injective mapping $f:\mathbb{N}^\mathbb{N}\rightarrow \{0,1\}^\mathbb{N}$ for which $\{0,1\}^\mathbb{N}\setminus f\left(\mathbb{N}^\mathbb{N}\right)$ is countable From 1. and 2. it follows directly that $\{0,1\}^\mathbb{N}\sim\mathbb{N}^\mathbb{N}$ Points 1. and 3. should be clear, let […]

Can one come to prove Cantor's theorem (existence of higher degree of infinities) FROM Russell's paradox?

I have been thinking about this: One can arrive at Russell’s paradox from Cantor’s argument, but can we go the other way round, i.e., can we prove Cantor’s diagonal argument(often referred to as Cantor’s paradox) from the conclusion of Russell’s paradox using the Axiom Schema of Specification/Sepration– there is no universal set. What do other […]

Sum of combinations of the n by consecutive k

In a book, I found that the sum of combinations: $\binom{n}{k} + \binom{n}{k+1} +\cdots+ \binom{n}{n}$, where k starts from 0, equals $2^n$. It is possible to express this statement via sum: $2 + \sum_{k=1}^{n-1}{n!\over k!(n-k)!} = 2^n$, using binomial theorem. I tried several small n values, such as 4, 6 and others, the statement looks […]

Unions and Functions on Sets

Given these conditions, I seek a proof. Let $f: A \rightarrow B$ be a function, and let $X$ and $Y$ be subsets of $A$. Prove that $f(X \cup Y) = f(X) \cup f(Y)$. I can’t seem to figure it out. It appears obvious, but materializing a proof is troubling me. What is the best method […]

What is the cardinality of the set of all functions from $\mathbb{Z} \to \mathbb{Z}$?

How can I approach this? I have to find the cardinality of the set of the functions from $\mathbb{Z} \to \mathbb{Z}$ and I have no idea on how to solve it. Can someone hint me here? The approach itself is what confuses me… how do I try to map this to something else since its […]

Defining additions in the ring of integers

Is it possible to describe all possible ways in which one can define additions in the set of integers to give it a structure of ring when the multiplication is same as the usual multiplication ? If the addition is same as the usual addition then one can easily describe all possible multiplications. What can […]

Show that 2S = S for all infinite sets

I am a little ashamed to ask such a simple question here, but how can I prove that for any infinite set, 2S (two copies of the same set) has the same cardinality as S? I can do this for the naturals and reals but do not know how to extend this to higher cardinalities.