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- What strategies can I follow to find the counterexample of an invalid categorical reasoning? July 15, 2024I am taking the discrete mathematics course in Engineering and I am having problems with the reasoning exercises in the logic part. I have an extremely hard time finding suitable propositional functions and a universal set that invalidates the reasoning, for example with these two invalid reasonings: ∀x: [d(x) ⇒ c(x)]; ∃x: [-c(x) ∧ p(x)] […]Lucio Mazzini
- Is there a paradox similar to Russell's paradox in the proof of Godel's incompleteness theorem? [closed] July 15, 2024The Gödel Incompleteness Theorem was a major discovery in modern logic that has consistently attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness Theorem was put forward, the scientific and philosophical significance of its proof has been questioned; in particular, Wittgenstein regarded it as a certain logical paradox. In Gödel’s view, […]Zhang Hong
- logic puzzle birth year July 15, 2024Suppose we have two people A and B. A died 129 years after B was born. At least one of A or B was alive for exactly 100 years. B died in 30 B.C. When was A born? This confused me. If B lived 100 years, B would be born in 130 BC and so […]james
- Prove $(B \implies (C \implies D)) \implies (C \implies (B \implies D))$ without the Deduction Theorem July 15, 2024I am reading "Introduction to Mathematical Logic" by Elliott Mendelson, and I am currently at the axiomization of propositional calculus. Mendelson presents the following three axioms (with modus ponens as the only rule of inference): $A1: B \implies (C \implies B)$ $A2: (B \implies (C \implies D)) \implies ((B \implies C) \implies (B \implies D))$ […]gestory2
- If sub-universe $S$ of lattice has congruence $\theta$, does the lattice have a congruence $\lambda = \theta \cap S^2$? [duplicate] July 15, 2024Let $(L, \lor , \land )$ be a lattice and $S$ a sub-universe of the lattice. A sub-universe of a lattice will be any subset of the lattice set that is non-empty and closed under $\land$ and $\lor$. Let $\theta$ a congruence of $(S, \lor_{S\times S}, \land_{S\times S})$. Is it true that there is a […]lafinur
- Is implication true if two statements are always the case? July 14, 2024I have a task that requires me to show that under a certain set of circumstances, a set has property A if and only if it has property B. I can show that under the given circumstances, the set always has property A. I can also show that it always has property B. Since these […]ormondo
- Examples of index set not Turing equivalent to the Halting Problem? July 14, 2024By definition, a set $I \subseteq \mathbb{N} $ is an index set if $\forall i,j ((i \in I \land \varphi_i = \varphi_j) \implies j \in I)$. Thanks to the Rice's Theorem, we know that, said $F$ a family of partial computable functions on the naturals, the set of their code $E = \{e \in \mathbb{N} […]NON
- Understanding the definition of congruences over a lattice July 14, 2024Let $(L, \land, \lor)$ a lattice and $\theta$ a binary relation over $L$. We say $\theta$ is a congruence iff $$ x_0\theta x_1, y_0 \theta y_1 \Rightarrow (x_0 \lor y_0)\theta(x_1 \lor y_1) $$ (and the same for $\land$). I am confused by the required cardinality of $\theta$. Any equivalence relation is isomorphic to a partition […]lafinur
- Which statement is not a mistake that Reina has made? July 13, 2024"A survey done at a certain high school found that any student who liked tennis also liked swimming. They also found that students only liked swimming if they could swim." Reina: If 30 students from the high school can swim, then 30 students from the school also like tennis. If the quoted paragraph above is […]Bacterigerm
- Number of lattices over a finite set July 12, 2024I'm interested in finding a general formula for the number of lattices possible over a finite set $S$ as a function of the set's cardinality. For instance, how many lattices over $\{1, 2, 3\}$ are there? Since the set contains three elements, and a lattice is a partial order, we must count the number of […]lafinur
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