Tag: calculus
Logic on Math StackExchange
- Iff propositions where both directions require choice? May 28, 2023Recently, I have been revising a basic course on noncommutative rings and modules over them. One proposition proven early on is all left modules over $R$ are free iff $R$ is a division ring and an interesting point is that, at least in the standard proofs we were given (and I have seen repeated in […]Isky Mathews
- Satisfiability in greater languages May 28, 2023Let $\mathcal{L}, \mathcal{L}'$ two languages in first order logic such that $\mathcal{L}\subset \mathcal{L}'$, $\Gamma \subseteq Form_{\mathcal{L}}$ and $\varphi \in Form_{\mathcal{L}}$. Prove that if $\Gamma \vdash_{\mathcal{L}'} \varphi$ then $\Gamma \vdash_{\mathcal{L}}\varphi$. In my book, there is a proof when $\mathcal{L}'$ is an extension of $\mathcal{L}$ with only constant symbols, but this is the general case. I am […]Superdivinidad
- How to solve $(x-1)(x-2)=0$ constructively? May 28, 2023I want to prove that $$(x-1)(x-2)=0\Leftrightarrow x=1, 2$$ $\Leftarrow$ is easy. The problem is $\Rightarrow$. Assuming $x\neq 1, 2$, we can derive $1=0$ by dividing both sides of $(x-1)(x-2)=0$ by $x-1$ and $x-2$. Thus we get $\lnot \lnot (x=1, 2)$. However, intuitionistic logic cannot eliminate double negation.BonBon
- How do we know we can "enumerate the primes" in this proof of the infinitude of primes? May 28, 2023In the presentation my course notes give of Euclid's proof, it is mentioned that we could enumerate the primes given that there is a finite amount of them. I have a few questions about just this part. What exactly is meant by "enumerate" here? Does it mean "ascertain the exact values of the primes", "count […]God
- Can a contradiction prove a contradiction like this? May 28, 2023If I prove the implication ¬P ⇒ ¬R∧R, and then I prove the implications ¬R ⇒ Q, R ⇒ ¬Q, is it valid to say ¬P ⇒ Q∧¬Q? I am unsure because while it is the case that Q and ¬Q both follow from P, it seems that we may have assumed that R is […]God
- About inference in mathematical induction of first-order predicate logic. May 28, 2023In induction of first order logic, reasoning with n∈ℕ : P(n)➞P(n+1). {n=1,P(1)} ⊨ P(1) k∈ℕ, {∃k: P(k)} ⊢ P(k+1) K:={ k | P(k), {P(k)} ⊢ P(k+1)} (then K⊂ℕ), {P(1) ∧ (P(1) → P(2))} ⊨ (P(2) ∧ 1∈K), {P(2) ∧ (P(2) → P(3))} ⊨ (P(3) ∧ 2∈K), ...... ∴∀n∈K, ⊨ (∀n: P(n) ∧ (P(n) → P(n+1))) […]DoP
- Does $(x\in A\land y\in B)$ follow from $xRy$? May 27, 2023I have defined “being in $R$-relation to” as: Given a binary relation $R\subseteq A\times B$ and the elements $x\in A$ and $y\in B,$ $$xRy\iff (x,y)\in R.$$ For $R$ to be injective, is $$(xRz\land yRz)\implies x=y$$ acceptable or is $$\forall x, y, z\;\big((x,y\in A\land z\in B \land xRz\land yRz)\implies x=y\big)$$ necessary? But doesn't $(x\in A\land y\in […]Leonardo Orietti Del Duca
- Existence of some syntactic deduction May 27, 2023Given an $\mathcal{L}-$language, prove if it exists or not a deduction for: $$\exists x_1 \exists x_2 \neg \varphi \vdash \neg \exists x_1 \exists x_2 \varphi $$ My idea is that if it exists a syntactic deduction, by the Soundness Theorem it follows the semantic implication, so a set composed of the first formula and the […]Superdivinidad
- Texts on the logic of chess May 27, 2023I am looking for texts that discuss the logic of the game of chess. I am sure there are a few such texts out there. Such a text might formalize chess in first-order logic. I would be very grateful if someone gave me a list of texts on the logic of chess.user107952
- If a set is arbitrary, can you prove something about it by constructing an example of it? May 27, 2023Let $R$ be a relation from $A$ to $B.$ Prove that $\operatorname{Domain}\left(R\right)\times \operatorname{Range}\left(R\right)\not\subset R.$ I need to show that there is some couple $(a,b)\in \operatorname{Domain}(R)\times \operatorname{Range}(R)$ such that $(a,b)\notin R.$ It is easy to show by example that the theorem is true: A={1,2} B={3,4} R={(1,3),(2,4)} Domain(R)={1,2} Range(R)={3,4} Domain(R) x Range(R)={(1,3),(1,4),(2,3),(2,4)} In this case, the theorem […]lightyourassonfire
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