# Tag: .999

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- Prove $(B \implies (C \implies D)) \implies (C \implies (B \implies D))$ 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 $\lambda = \theta \cap S^2$? 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
- Clock that is twice as fast July 14, 2024I have a grandfather clock that runs twice as fast as a normal clock. Can I still read it somehow? I've tried a few things out of desperation, like dividing by 2. But it was clear that that wouldn't workRedcommander
- 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
- Is there a propositional proof system that is not known to be simulated by extended Frege? [closed] July 12, 2024Extended Frege is a propositional proof system that is achieved by adding the extension rule to a Frege system. That rule allows to replace formulas with fresh variables. A propositional proof system $P$ is said to simulate a propositional proof system $Q$ if for every proof in $Q$ of length $|x|$, $P$ has a proof […]rus9384
- Proving the Negation of a Formula does not Require the Formula as an Assumption July 12, 2024The following lemma states that if we can prove negation of a Well Formed Formula (WFF) $\alpha$ by assuming the formula itself, then we can do it without such an assumption. Lemma. Let $\Sigma$ be a set of WFFs and $\alpha$ a WFF. If $\Sigma \cup \{\alpha\} \vdash (\neg \alpha)$, then $\Sigma \vdash (\neg \alpha)$. […]Hosein Rahnama

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