Recent content by Andrei1

The set \[ \{s\} \] has power set \[\{\varnothing, \{s\}\}.\] Empty set is proper by the definition, but it is not yet introduced. So we don’t know if \[\{s\}\] has any proper subsets.

I suggest to prove the following: $$\forall \epsilon>0\exists\delta>0\forall x\in\mathbb{R}^*\left(|x|<\delta\to\left|\frac{|2x-1|-|2x+1|}{x}+4\right|<\epsilon\right)$$ which defines $$\lim_{x\to 0}\frac{|2x-1|-|2x+1|}{x}=-4.$$

Part (b) is not a sentence to prove because $$x$$ is free. (b) Explain why there can be no value of $$\delta>0$$ such that for all $$x\in [0,\ 2]$$ if $$0<|x-1|<\delta$$, then $$|f(x)-4|<1$$. Now this is a sentence to prove.

About (b). Let's take a 4-clique. Any subgraph of it is not a substructure if it is not a clique. So I have, up to isomorphism, 4 substrucutures: one 1-clique, one 2-clique, ...

My answers: (a) $$2^n$$ (b) $$n$$ (c) $$2^n$$ Are they correct?

For example, the division algorithm theorem has the form $$\forall m\in\mathbb{N}^*\forall n\in\mathbb{N}A(n,\ m)$$. Here we let $$m$$ be some arbitrary positive natural number and then we can prove $$\forall n\in\mathbb{N}A(n,\ m)$$ by induction on $$n$$ (strong induction).

Here is a problem from Zorich's "Mathematical analysis I", pg.69. I suspect this text has misprints: is it correct that $$n=1$$ under $$\sum$$ and why, or it should be $$n=0$$? By order I understand the unique $$p\in\mathbb{Z}$$ such that $$q^{p}\leqslant x<q^{p+1}.$$

Here's the plan. (1) Prove $$a\subset b^+\Rightarrow b\notin a$$ by induction on $$a.$$ Use also $$x=y\Rightarrow x^+=y^+.$$ (2) Prove $$a\subseteq b\Leftrightarrow a\subset b^+$$. In proving ($$\Leftarrow$$) side use (1). In proving ($$\Rightarrow$$) side use $$x\subset x^+$$, which follows...

Prove that for all $$x,y\in\omega,\ \ x\subset y\vee y\subset x.$$ If I assume that the conclusion is false then I can prove that for some $$a\in x,\ b\in y$$ we have $$a\notin b$$ and $$b\notin a.$$ Also I am thinking that if assume the contrary then $$\omega$$ minus $$\{x\}$$ or minus...

I took this problem from Zorich "Mathematical analysis". Zorich spoke just twice about infinity before this problem: 1. Dedekind's definition, 2. Axiom of infinity. Suppose there is a bijection from $$x$$ to $$x^+.$$ Since $$x\subset x^+$$, then $$x$$ is infinite by 1, which is false, but I...

Let $$x$$ be a natural number (set). How to prove that there is no bijection between $$x$$ and $$x^+$$, where $$x^+=x\cup\{x\}$$? Then I can show that $$\mathrm{card}\,x<\mathrm{card}\,x^+.$$ I know that $$x\notin x.$$

In his book Hedman uses a shorter form of induction. For that he shows the property of the formula is preserved under equivalence: if the embedding preserves $$\psi$$ and $$\varphi\equiv\psi$$, then the embedding preserves $$\varphi.$$ How to prove this? Since $$\varphi$$ and $$\psi$$ are...

I didn't formulate exactly my question. I just need to read somewhere the proof of the theorem that literal embeddings preserve quantifier-free formulas. What good sources do you know, excepting Hedman's book?

I want to understand the notion of equivalence. $$\varphi\equiv\psi$$ means that for all structure $$M$$, $$M\models\varphi$$ iff $$M\models\psi.$$ Suppose that for some structure $$M$$ and formula $$\varphi$$, $$M\models\varphi(\bar{a})$$, where $$\bar{a}$$ are all constants in $$\varphi.$$...

$$T=918\pi$$ is the least common multiple of those periods. I found this. But why it is the smallest positive period of $$f$$? For example, I consider the functions $$f_1(x)=\sin x$$ and $$f_2(x)=\operatorname{tg} x-\sin x$$, which both have $$2\pi$$ as main period. But then $$\pi$$ is the main...