Tag: OpenLab8
Logic on Math StackExchange
- Can a Program Certainly Distinguish Normal Distribution from a Discrete Finite One October 3, 2024Suppose we have two distributions $\mathcal{A}$ and $\mathcal{B}$. One of the distributions is normal $\mathcal{N}(0,1)$ and one is a discrete distribution with finite support, i.e. can be represented as a finite sum of weighted delta measures $\sum_{k = 1}^n c_k\cdot\delta_{b_k}$, but we don't know which one is which. Now a natural question to ask if […]Sergey Novozhilov
- A statement assumes that P is true and asks to prove that Q is false. What can we do to achieve this? October 3, 2024which of the following propositions are correct? Suppose that P is true and demonstrate that Q is false. Suppose that P is false and demonstrate that Q is true. Suppose that Q is false and demonstrate that P is true. Suppose that Q is true and demonstrate that P is false. Suppose that P and […]Evelyne Gousseau
- Is $A $ implying $B$ really captured by $A \implies B$? [closed] October 2, 2024Consider for example a common statement for a function f from basic calculus : $$ \text{differentiability of (f)} \implies \text{continuity of (f)} -(1)$$ Now, if we have a function is discontinuous, then it would be an acceptable deduction (... for most math students) to deduce from the above that it can not be differentiable.(*) But, […]Cantor Dust Drachen
- two issues on first order logic's GEN rule October 2, 2024The most popular axiom system for first order logic contains 5 axioms and 2 rules,the rules of inference of any first-order theory are: 1 Modus ponens(MP rule): C follows from B and B → C 2 Generalization(GEN rule): (∀x)B follows from B In the book Introduction to Mathematical Logic (MENDELSON 6th Ed - CRC Press) […]showkey
- Strength of Axiom of Choice vs. Law of Excluded Middle vs. Dependent Choice October 1, 2024tl;dr Are there results comparing how many results in ZF can be proven with axiom of dependent choice (DC) vs. law of excluded middle (LEM) vs. axiom of choice (AC)? To hopefully nip in the bud any issues of ambiguity, let me clarify from the outset that I am speaking in terms of constructive set […]DiracComb16796
- Issue with contrapositive October 1, 2024So contrapositives are logically equivalent statements, which can be useful when proving things. But I'm not convinced that this is true. Take this example: $x \in \mathbb{Z}$. Then $$x \neq 2 \Rightarrow x+1 \neq 0.5$$ is true, but the contrapositive $$x+1 = 0.5 \Rightarrow x=2$$ is false. If one statement is always false/true, then is […]thedanktouch
- How can we know that a model is a valid model for a given set of axioms? October 1, 2024For a while I have thought (naively) that maths is constructed in the following way: pick a set of axioms and some rules of logic, then deduce all the theorems you can about it. The goal being obviously to have the simplest set of axioms (and simplest logic rules?) such that you recover the mathematical […]Flavien Hirsch
- Does universally common truth value mean that two predicates can be turned into each other by logic rules? October 1, 2024Consider two predicates, each made of $n$ variables. If they have the same truth value for every evaluation, must they be the same predicate? By same predicate, I mean that both can be simplified into a common form, using the simplification rules of logic.Cantor Dust Drachen
- In what ways the consistency of ZF can be proved? October 1, 2024Recall that whether ZF is consistent is essentially a combinatorial problem about a string rewriting system: We define a set of symbols $S$ to write with, and the subset of all strings $s \in S^{\star}$ that are well-formed formula. Then we define a certain set of axioms (each is a well-formed formula), and a set […]Student
- How do I show that $(p \lor q) \land \{\lnot [(\lnot p \land r) \lor q] \}$ is tautology by showing the axiomatic proof? [closed] September 30, 2024I'm getting a bit confused, I was told to answer this but when I try to do it. The expression does not hold true for all truth values of $p,q,$ and $r$. Making $(p \lor q) \land \{\lnot [(\lnot p \land r) \lor q] \}$ not a tautology.Ethaniel Manuel
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