Tag: .999
Logic on Math StackExchange
- Axioms as PARTIAL information givers of primitive terms - Enderton's Elements of Set theory September 24, 2023This is an excerpt from Elements of Set theory by Enderton.(p. 11) Our axiom system begins with two primitive notions, the concepts of "set" and "member." In terms of these concepts, we will define others, but the primitive notions remain undefined. Instead, we adopt a list of axioms concerning primitive notions. (The axioms can be […]Harshit Rajput
- How to show that $p∨q, q∨r, p\to ¬r ⊢ q$? September 24, 2023How to prove $p∨q, q∨r, p\to ¬r ⊢ q$ using natural deduction? I don't really know how to go about it since I have 2 or statements, but here is my try. $p∨q$ premise $q∨r$ premise $p\to ¬r$ premise $p$ Assumption 1 $¬r$ $\to e 3,4$ $q$ Assumption 2 I am kinda lost and don't […]User
- How to determine all real numbers such that$ \forall n\in \mathbb{N},\:x^{n+2}\:\le \:x^{n+1}+x^n$ [closed] September 24, 2023I've just started higher education in France in what is called "preparatory classes for engineering schools", and the first chapter that we studied is about logic in discrete mathematics. This is the first time I'm studying discrete mathematics and I find it very abstract, too abstact even (keep in mind I'm only 18 and I've […]Sherif Robov
- Understanding indexed families of sets September 24, 2023So I am looking at this family of sets. It has an indexed set of $[0,1]$ where every $x∈[0,1]= \{y∈R |x_0≠0\ \text{AND} \ x_0≤y≤1 \}$ The question is asking which of the following six statements are true. But I am not interested in the answer, I just want to understand what exactly the indexed set […]Adam R. Johnson
- Understanding logic formulae involving $P(n)$ September 24, 2023I am given that that $n,m,$ and $k$ are positive natural numbers, and have to find out which of the six statements above are true. I know that $P(1)$ is not true, because $1$ cannot be written as $5k,$ where $k$ is a natural number; so, I can rule out the first possibility. But I […]Adam R. Johnson
- Prove: |= ∀xϕ→ψ ↔ ∃x(ϕ→ψ), with x not belonging to FV (ψ) September 24, 2023|= (∀xϕ→ψ) ↔ (∃x(ϕ→ψ)), with x not belonging to FV(ψ) (Free Variable). I have been struggling with this exercise for a while. I have to prove this formula using either the valuations or as shown in this example in Van Dalen's "Logic and Structure" (I do not know the name of this method) with a […]MatP25
- What is the justification for the RAA rule, and how does it relate to the Principle of Explosion? (Mathematical Logic by Chiswell and Hodges) September 24, 2023I am reading through Mathematical Logic by Ian Chiswell and Wilfrid Hodges and am trying trying to finalize my understanding of the RAA rule introduced in section 2.6. The book states the rule in the following way: Suppose we have a derivation whose conclusion is $\bot$. Then there is a derivation $$\require{cancel}(\cancel{~\lnot\phi~}) \\\\\ D \\\\\ […]Artyom Elessar
- Universal generalization and the Deduction theorem September 24, 2023Universal generalization The rule of universal generalization in FOL is stated as follows: (Gen) From $p$ we may infer $(\forall x)p$ provided $x$ does not occur free in any premiss which has been used in the proof of $p$. I don't understand what this means, particularly for deduction sequences. Given a set of formulae $S$ […]Hernán Ibarra Mejia
- Gödel Incompleteness Theorems Proof September 23, 2023Is the proof outline of Godel’s Incompleteness Theorems on Simple Wikipedia correct?Caleb
- Tautological statements September 23, 2023I have been tasked with finding how many of the following statements are tautologies, and was wondering which method would be the best to solve this assignment fast. My teacher said I should use no more than 5 minutes on it. How can I quickly spot which of these statements are tautologies?Marcus K. Johnson
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