Grades can be found in “Dashboard / OpenLab Gradebook” – let me know if you have any trouble viewing them. Exams will be returned in class on Tuesday.
See you then,
Prof. Reitz
Logic on Math StackExchange
- Verify if my proof that S ◦ R = ∅ iff Ran(R) and Dom(S) are disjoint is valid December 21, 2024I came across this problem from Velleman's How To Prove It. Suppose R is a relation from A to B and S is a relation from B to C. Prove that S ◦ R = ∅ iff Ran(R) and Dom(S) are disjoint. This is my initial proof although upon checking the solution, the author used […]APGG_B08
- How to prove that F∧φ is unsatisifiable doesn't mean F is unsatisfiable [duplicate] December 21, 2024The statement can be proven by contradiction. For instance, we can directly set $\phi$ as a contradictory formula and $F$ as a tautology. However, if we want to argue this point positively, how should we proceed? Here are my thoughts： Since $F \land \phi$ is unsatisfiable, it means that for any assignment $v$, we have […]yXin Huang
- how do I write this proof in natural deduction? [closed] December 21, 2024((A ∨ B) → C), ((C ∨ D) → F) ⊢ (A → (E ∨ F)) (A / B) -> C :PR (C / D) -> F :PRisa
- Hilbert calculus and ZFC and their relation to independent statements [closed] December 20, 2024I have recently been studying calculi, namely the Hilbert calculus or ZFC. To conclude, I am making some general remarks in my notes on consistency: In axiom systems you cannot simply infer from $\Sigma \nvdash \Psi$ to $\Sigma \vdash \neg \Psi$, as there can exist formulas $\Psi$ independent of $\Sigma$ for which $\Sigma \nvdash \Psi$ […]God
- What's wrong with my formal translation of "every positive number has exactly two square roots"? December 20, 2024I translated the statement "every positive number has exactly two square roots" as $$\forall x, x>0\implies (\exists!y,\exists!z,yKenneth Luo
- How would you translate this statement into formal logic? [closed] December 20, 2024Let $N$ be the product of two prime factors $p$ and $q$. For all $g$ positive integers, $m$ positive integers, there exists a positive integer $r$ such that: $g^r = mN + 1$Filip Rank
- Predicate Logic: Is there a particular reason for using proof of contradiction here? December 19, 2024In Logic in Computer Science, the solution to 2.3.9 (d) is as follows : https://i.sstatic.net/TMR5LzRJ.png Can this question be proved without using contradiction, as shown below? \begin{array}{} 1 \quad \forall x P(x) \to S \quad \text{prem} \\ ------------\\ 2 \quad x_0 \quad P(x_0)\to S \quad \forall x e\ 1 \\ 3 \quad \exists x (P(x) […]RuntimeErrorz
- Logical expression of Neighborhood axioms [closed] December 19, 2024While taking notes about neighborhood axioms for definition of topological space on proofwiki, I had some confusion regarding logical expressions. Question 1: In the original page, the writer used ":" (possibly to avoid double use of ","). Is that a common convention? Even when I understand them as "such that", it doesn't make much sense […]Unuu U
- Application of "texts" in a system [closed] December 19, 2024Initial Opportunities for Reasoning About Texts The following initial opportunities are assumed for a mathematician reasoning about texts: The opportunity to insert one text $\Phi$ into another text $\Psi$ (in particular, to write alongside), inserting every occurrence $\Omega$ in the text $\Phi$ in the capacity of an occurrence in the text $\Psi$. The opportunity to […]MDNFromVN
- Spectrum of a Satisfiable Sentence December 19, 2024Let $\mathcal{L}$ be a first order language including only the equality sign $\approx$. Let $A$ be an arbitrary sentence in $\mathcal{L}$ and define its spectrum as follows $$\text{Spec} A = \{\text{Card} |\mathfrak{M}|: \, \mathfrak{M} \vDash A, \, \text{Card} |\mathfrak{M}| < \aleph_0\}.$$ For every $n\in\mathbb{N}$, let $\lambda_n$ be a sentence saying that "There is at least […]Hosein Rahnama
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