Logic Definition and 1000 Threads

Hello, I have a few questions and I'd appreciate if you can please help me. 1. If I want to say "for every ## i \in \Bbb N ## and ## 0 \leq j \leq i ## define ## A_{i,j} := i ## and ## B_{i,j} := i \cdot j ## ", then is the logical formula used for this is as such?: ## \forall i \in \Bbb N...

Can anybody check my work regarding these three statements, the third, in particular, please? I think I got the first two statements down, but I think that I'd feel safer if I got a second opinion. I think I also have a correct translation of the third statement down, but only because I...

Just wanted to share a cool proposal paper from the BASE collaboration. I found this article dense and the theoretical aspects are way above my pay grade, so please chime in if you think I get anything wrong. For the record, I have no ties to this group and hadn't heard of them before this...

Problem: Let ## f: \Bbb R \to \Bbb R ## be continuous. It is known that ## \lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \} ##. Prove that ## f ## gets maximum or minimum on ## \Bbb R ##. Proof: First we'll regard the case ## l = \infty ## ( the case...

I hope someone can help me or point me in the right direction. I am reading Discrete Mathematics with its Applications by Rosen. I am trying to self learn discrete math. I am actually able to do most questions but I have a question about a solution (not the question itself.) The question is...

Not sure if this is an allowed post, as it is not technically math but I'm trying to work through the below proof. If workers have a fundamental right to a job, then unemployment will be virtually nonexistent but job redundancy will become a problem. If workers have no fundamental right to a...

I failed to become a mechanical engineer because I could not learn how to do sequences and series in Calculus II. I could get Cs and Bs on all the concepts of Calculus II until I got to Sequences and Series. Then I would get F minuses on any tests involving series problems such as Infinite...

I obtained the following result: ([(A xor B) xor 0]* AC)' ([(A'B + AB') xor (0)]*AC)' [([A'B + AB']*(0)' + 0)*AC)]' [(A'B + AB')*(1)*(AC)]' [A'ABC + AB'AC]' [AB'C]' A'+B+C' the solution to this problem is getting a different answer, I don't know why this solution isn't inverting the output AB'C

Ok, so assume we have a model for ZFC where CH does not hold. What values may ##2^{\aleph_0}## assume over said models?

Upper bound definition for sets: $ M \in \mathbb{R} $ is an upper bound of set $ A $ if $ \forall \alpha\in A. \alpha \leq M$ Upper bound definition for sequences: $ M \in \mathbb{R} $ is an upper bound of sequence $ (a_n)$ if $ \forall n \in \mathbb{N}. a_n \leq M$ Suppose we look at the...

An example we were given is as follows: {ua|u∈∑*} (where ∑* is set of all words over ∑) so we have ∀x. last(x) → a(x). I am given {awa|w∈∑*} to do, and I know that I have to express that a is the first letter and last letter in a word. Could I write it as: ∀x,y ( a(x) ∧ a(y) ∧ x<y → ∃z(x<z<y))...

I can't figure out this: Say we have event B given α, denoted as {B|α}. If B happens, that implies that R happens: {B|α} → R. Now I want to apply modus Tollens. So if I do, do I get the result: ¬R → {¬B|α}? I mean, I hope I can keep the α unaffected. Is that the case? ¬X meaning X does not happen.

In the book: SET THEORY AND LOGIC By ROBERT S.STOLL in page 19 the following theorem ,No 5.2 in the book ,is given: If,for all A, AUB=A ,then B=0 IS that true or false If false give a counter example If true give a proof

In Tao's Analysis 1, Lemma 5.3.6, he claims that "We know that ##(a_n)_{n=1}^{\infty}## is eventually ##\delta##-steady for everyvalue of ##\delta>0##. This implies that it is not only ##\epsilon##-steady, ##\forall\epsilon>0##, but also ##\epsilon/ 2##-steady." My question is, why do we need...

##A \subset B## means that ##\exists x \in B ## such that ##x \not\in A##. Is this logically equivalent to ##\exists x \in B \land x \not\in A##? Formally, $$A \subset B \iff {(\exists x \in B \land x \not\in A)\land(\forall y \in A \land y\in B)}$$ I have tried consulting...

Details of Question: ds/dt= v which becomes ds=v dt, where s=displacement, t =time, and v=velocity Then we can integrate both sides of this equation, and do a little algebra, and turn the above equation into: s − s0 = v0t + ½at2 My main question is about the integration of...

Can anyone advise me on a good introductory website about (formal) logic? Preferably about logic in a broad sense.

Checking my understanding. Can it be said that it is the overlap of the reach and effect of each of the 4 forces, from each respective point of origin possible within the universe, that gives us universal general relativity? Like the most intricate gear set ever? Could it then also be said...

Given that the negation is distributed across parenthesis, P become ~p and S gets double negation ~~S. Hence my solution was " I will not buy the pants but I will buy the shirt. (or and I will buy the shirt, since but can be used in the place of and). This is from How to prove things by...

Summary:: . When asked to prove by Induction, I'm asked to prove a statement of the form: Prove that for all natural numbers ##n##, ## P(n) ## Which means to prove: ## \forall n ( P(n) ) ## ( suppose the universe of discourse is all the natural numbers ) Then, I see people translating...

Suppose I have the following ( arbitrary ) statement: $$ \forall x\in{S} \ ( P(x) ) $$ Which means: For all x that belongs to S such that P(x). Can I write it as the following so that they are equivalent? ( although it is not conventional ): $$ \forall x\in{S} \land ( P(x) ) $$ Can I write...

Logic and equations seem to have come out of nowhere in this question. I have been unable to understand where these equations come from and why they are used. Can someone describe the logic for the steps in the question?

A book that has really caught my attention recently is Yuri Manin's A Course in Mathematical Logic for Mathematicians. I am very interested in the foundations of mathematics and mathematical logic, plus I noticed that it had some chapters on quantum logic, so I started skimming through it...

A computer language is not a clear example of a formal language, a formal system, or a formal logic. Can we modify the rules for a computer language to create clear examples? My non-authoritative classification of topics in mathematical logc is: 1. Formal languages. A formal language is...

I recently encountered mirror symmetry method of solving circuits and by it solving circuits became very easy but problem I am facing with it is that I can't figure out logic behind it. For example if we try to simplify this circuit Then we say that if ##I## current flows from Point A to C then...

Hello, I am a retired Marine that has decided to pursue an undergraduate degree in programming and am currently working on paper that discusses the future of Mobile Computing. In this paper I am introducing a hypothetical process that assists in forecasting the future of mobile computing. To do...

Given: x\in A\cap B\leftrightarrow x\in A\wedge x\in B x\in A\cup B\leftrightarrow x\in A\vee x\in B x\in A-B\leftrightarrow x\in A\wedge x\notin B A=B\leftrightarrow(\forall x(x\in A\leftrightarrow x\in B)) Then prove using only the above and the laws of logic that: ™ (A\cup B)-(A\cap...

If someone is lying, who copied the assignment? Alex: Cate copied the assignment. Cate: David copied the assignment. David: Cate is lying. Keil: I didn't copy.I think Cate is lying. If Alex is true and there is only one person who is lying, Cate and David can't be true at the same time. If...

One proposal that I have read (but cannot re-find the source, sorry) was to identify a truth value for a proposition (event) with the collection of closed subspaces in which the event had a probability of 1. But as I understand it, a Hilbert space is a framework which, unless trivial, keeps...

[mentor note: edited for clarity] The man weighs 100kg. All constructions, ropes, universal joints, rollers, fans, etc are massless. Friction between weight scale and feet or construction is enough to hold side forces... The red line is the rope. What will the weight scale show? <100kg...

Hi all, Having some problems digesting on electric circuits. Below is an example of a question and I would like to ask how do I go ahead in solving this. Firstly, for these types of questions: I have understood how to write a function table, and it goes something like this: Now what I am...

Sir/madam, I request you to solve 2 questions ( q-3 and q-5 ) of symbolic logic ( Strenthened method of conditional proof ). These questions are taken from I.M.Copi's 'symbolic logic' ( edition -5, sec. 3.8, pg- 61 ) File is being attached. thank you yours truly Deep Kumar Trivedi

Basically the problem starts with these given premises: 1. ~ (A ∨ (B⊃T)) 2. (A ⋅ C) ∨ (W ⊃ ~D) 3. ~(P ∨ T) ⊃ D 4. ~P ≡ ~(T ⋅ S) And from these premises, I must prove ∴ ~W. This is what I have done so far: 5. (~P ⊃ ~(T ⋅ S)) ⋅ (~(T ⋅ S) ⊃ ~P) B.E. 4 6. ~P ⊃ ~(T ⋅ S)...

As in the summary: is the term "theory" only used for binary-valued logics? If not, how is it defined for multi-valued logics?

I want to get started with FOL and decided to get through some very basic book first. Currently looking at: A Modern Formal Logic Primer - Teller An Introduction to Formal Logic - P.Smith Logic: The Laws of Truth - J.J. Smith These 3 books are frequently recommended I just don't know which...

Nobel laureate Hans Bethe was a friend of mathematician-physicist John von Neumann, and he once said: "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man" and "[von Neumann's] brain indicated a new species, an evolution beyond man"...

$\left\{ \begin{array}{rcl} x+y+z &=& 1\\ x^2+y^2+z^2 &=& 2\\ x^3+y^3+z^3 &=& 3 \\ x^5+y^5+z^5 &=& ? \end{array}\right.$How to find out $x^5+y^5+z^5=?$

i came acroos the below while studying propositional Logic, can anyone find the proofs 1) P ⊢ P 2) P → Q, Q→R ⊢ P → R 3) P → Q, Q→R, ¬R ⊢ ¬P 4) Q→R ⊢ (PvQ) → (PvR) 5) P →Q ⊢ (P&R) → (Q&R)

The paper https://www.whitman.edu/Documents/Academics/Mathematics/klipfel.pdf (beginning page 2``1) describes a model for experiments based (it says) on the book An Introduction To Hilbert Space and Quantum Logic https://www.amazon.com/gp/product/1461388430/?tag=pfamazon01-20. This approach...

One of the ways we use the term possibility is "We can go to Mars". Another way is "We may learn how to travel to Mars faster than light as our understanding of physics progresses". What exactly does it mean in modal logic? Would a statement like 1+1=2 be considered possible in modal logic or...

There are an infinite number of natural numbers. Why is that? Well this follows from the following facts: (i) There is at least one natural number. (ii) For each natural number there is a distinct number which is its successor, i.e., for each number $x$ there is a distinct number $y$ such that...

Trouble working through Set theory, Logic, and their Limitations by Maurice Machover. Particularly these 1. $\sigma \vDash \alpha \rightarrow \forall x\alpha$ where $x$ does not occur in a free $\alpha$ 2. $\sigma \vDash s_1 = t_1 \rightarrow ... \rightarrow s_n = t_n \rightarrow...

Hello! I'm a beginner in discrete math and don't actually know how to solve the bool algebra and logc problems. Sorry for errors in formulas - it's my first post here. I have some tasks that I want someone could help for me to solve. Step-by-step solutions would be really good, to really know...

If I go to town Iwill visit chris or john. But i own to chris 3 dollars and 5 to john.Hence if I visit them i have to pay my depts. So if I go to town how much money do i have to spend Whatever is your answer prove it

Summary: What did Omnès mean with this? I found an old article by Roland Omnès which analyzes the EPR paradox and offers a solution to it (https://www.sciencedirect.com/science/article/abs/pii/0375960189900182). At some point, the article says: "Some macroscopic systems do not satisfy the...

Summary: Advice on books for software engineer logic puzzles for interviews and improve overall skills I am looking to getting into software engineering, and am having trouble with find the logic puzzle books or websites to prep for interviews. Now of course there lots of logic puzzle type...

Which part of Newton's second law is definition and which part is law content?It seem that there is a violation in logic, because we define the notion of force through the notion of mass, then we define the notion of mass through the notion of force when we consider the second law.

Physicist Craig Hogan has proposed that the universe is based on holographic principle. To prove that the universe is a "hologram" he (and other physicists) have designed an experiment named "The Holometer" to measure quantum fluctuations that would become fuzzy at Planck scale...

Goldrei's Propositional and Predicate Calculus states, in page 13: "The countable union of countable sets is countable (...) This result is needed to prove our major result, the completeness theorem in Chapter 5. It depends on a principle called the axiom of choice." In other words: the most...

Goldrei's Propositional and Predicate Calculus states (in my words; any mistake is mine) that first-order logic is complete, i.e. any logic deduction from a set of axioms (written in first-order logic) is equivalent to proving the theorem for all models satisfying the axioms. Completeness is...