Questions concerning predicate calculus, i.e. the logic of quantifiers.

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2
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1answer
37 views

Deciding between implication and conjunction

This is one of the solved problems in Velleman's How to prove book: Analyze the logical forms of the following statements: 1) John likes exactly one person. ...
2
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4answers
74 views

Analyzing the logical form of “All married couples fight”

This is one of the example problems in Velleman's How to Prove book: Analyze the logical forms of the following statements. All married couples have ...
0
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2answers
43 views

Concluding Truth Value from Universe of Discourse

I have been working on the following problem from Velleman's How to Prove book: Are these statements true or false? The universe of discourse is the set of ...
3
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3answers
435 views

Idiomatic mathematical english statement for ∃x[P(x) ∧ ∀y(P(y) → y ≤ x)]

I have been working on problems from Velleman's How to Prove book and hit upon the following problem: Translate the following statements into idiomatic ...
-2
votes
1answer
71 views

Single best example for mathematical reasoning [closed]

Please provide the best example to learn/teach mathematical reasoning. It is suggestible that the problem should contain at least 3 mathematical statements (not core mathematical but like Socrates ...
1
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0answers
55 views

Doing a “harder” limit question from Stewart's Calculus using some predicate logic. Does my method work?

I am trying to come up with a method that simply allows me to calculate the $\delta-\epsilon$ proof of a limit, as long as the function being "limited" is invertible (bijective). As a proof of ...
0
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1answer
51 views

Have I proven this simple limit correctly? (trying a method that is new to me, using predicate calculus…but not yet “watertight”)

$\lim \limits_{x \to 3} x^2 = 9$ Very simple limit, but I am trying a "new" (for me) method: by definition: $\left(\lim \limits_{x \to 3} x^2 = 9\right) \quad = \left(\lim \limits_{x \to 3^+} x^2 = ...
0
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2answers
31 views

How to convert a sentence into first order logic?

There is a student who does not like punctual students. Where, S(x) = “x is student” P(x) = “x is always punctual” L(x, y) = “x likes y” can please help me? Thanks Alot~~
0
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1answer
32 views

Translating to English: $\forall x ¬(\forall y P(x, y)) \rightarrow \forall x \forall y ¬P(x, y)$

$$\forall x ¬(\forall y P(x, y)) \rightarrow \forall x \forall y ¬P(x, y)$$ I'm trying to intuitively understand this idea by thinking about it in terms of English. The second half is easy. Where P ...
0
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0answers
17 views

Analogue of Herbrand Disjunction for Negative Side of the Clark Completion?

For Horn clauses there is the following result. If T is a set of Horn clauses and p is a predicate, and if p is an existential consequence of T: ...
1
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1answer
101 views

Where do the topics covered in Lewis Carroll's 1896 book “Symbolic logic” fit in the modern mathematical curriculum?

Where do the topics covered in Lewis Carroll's 1896 book "Symbolic logic" fit in the modern mathematical curriculum? And what is the modern substitute or notation? It appears to me that all it covers ...
2
votes
2answers
45 views

What is the position of “exclusive or” in order of precedence for logical connectives?

In propositional logic the order of precedence I have found for the logical connectives is $\neg$ $\land$ $\lor$ $\Rightarrow$ $\Leftrightarrow$ Where do I have to put the exclusive or $\dot\lor$ ...
4
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1answer
33 views

Using expressions like $ \langle x,y \rangle$ in predicate logic formulas

I don't like how books on set theory write logic formulas when describing complex sets. For example that is how a regular book can show that some set $s$ is not a pair: $$\forall x \forall y (\langle ...
1
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1answer
43 views

Conjuctive Normal Form

In Boolean logic, a formula is in conjunctive normal form or clausal normal form if it is a conjunction of clauses, where a clause is a disjunction of literals; otherwise put, it is an AND of ORs. I ...
3
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1answer
33 views

Two-place position predicate problem

I see this sentence in one Logic Note Tutorial. What arguments are involved in any situation is determined by the meaning of the predicate. Sleeping can only involve one argument, whereas placing ...
-2
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1answer
51 views

Logic Substitution Problem

I see this formula on Logic Text Book, I take a picture and insert it here in order to one expert help me and correct the error. t is a term and $\varphi$ is a formula. I think one of ...
0
votes
1answer
32 views

Logic & Stucture & axiomatized [closed]

suppose L={C,B} be a First order language in which C and B is a single axiomatized predicate. if A be a set that has 3 elements, how many way we can convert A into a Structure for L? I try to solve ...
1
vote
1answer
58 views

Can universal instantiation be used more than once?

I'm trying to follow a proof in a logic text and it seems like the author used universal instantiation twice to reach the needed result. I was under the impression that you could only use UI one time ...
0
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0answers
23 views

proof of quantifier elimination in theory of real closed field/reals and existential quantifier over atomic formula

Standard proof of quantifier elimination for theory of real closed field/reals uses induction, as in Wikipedia article (http://en.wikipedia.org/wiki/Quantifier_elimination#Basic_ideas). However, it ...
1
vote
2answers
64 views

Represent “No naive is bad” using the first order logic.

How can I represent the following sentence using the first order logic? "No naive is bad" I had thought: $$\neg Naive(x)\vee Bad(x)$$
0
votes
1answer
66 views

How do I notate that $1/2$ children do not exist

Suppose that, the average person has $2$ $1/2$ children. only whole children exist It should be straightforward to notate: If an average person exists, then that person has two $1/2$ children ...
0
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2answers
72 views

Logic Inference Challenge [closed]

I read some logic course recently, would you please anyone say my inference is True? $\forall x S(x) \to \exists y(R(y)) \Rightarrow \forall x \exists y(S(x) \to R(y))$.
0
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1answer
35 views

proving{$\neg(\forall x)\alpha \rightarrow \alpha$}$\models$$(\forall x)\alpha$

prove {$\neg(\forall x)\alpha \rightarrow \alpha$}$\vdash\space(\forall x)\alpha$ Im not sure what is the convention, so to be clear I am talking about proving the formula from the seven axiom ...
-4
votes
1answer
157 views

Primitive Recursive Predicate Problem [closed]

i get trouble with 2011 midterm exam question. if P(x) and Q(x) be a primitive recursive predicate. which of the following is not a primitive recursive? anyone could describe it for me? 1) $P(x) ...
2
votes
1answer
20 views

find a sentence $\alpha$ in some language L such that

Let $K=\{k\in\mathbb N : k\mod2\not=0$ and $k\mod3\not=0\}$ find a sentence $\alpha$ in some language L such that $K=${$n\in\mathbb N :$exists a structure $M$ such that $M\models\alpha$ and ...
3
votes
2answers
80 views

$\forall$ At the beginning or at the end?

I have a set of real numbers $x_1, x_2, \ldots, x_n$ and two functions $f:\mathbb{R} \rightarrow \mathbb{R}$ and $g:\mathbb{R} \rightarrow \mathbb{R}$. What are the differences between the following ...
-1
votes
1answer
47 views

prenex equivalence problem

Suppose: $$\forall x\exists y \phi(x,y) \to \neg \exists x\psi(x) $$ which of the following formula are prenex normal equivalence with the above formula? i didn't any idea to explain it. it's a ...
0
votes
1answer
50 views

sentence in predicate logic

“If all politicians are showmen and no showman is sincere then some politicians are insincere.” Ans: F:= $(\forall x\,(P(x) \to ShMan(x)) \land \not \exists y\,( ShMan(y) \to Sinc(y))) \to \exists ...
0
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2answers
78 views

prenex normal equivalence challenges in math

consider these two following formula are prenex normal equivalence with the above formula? i think yes, but didn't have any idea to explain it.
-3
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2answers
65 views

check the formulas for satisfiability and validity [closed]

Could perhaps somebody explain me how to define if a formula is valid, satisfiable, or not satisfiable if ...
0
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1answer
32 views

Hierarchy of operators of propositional logic

I have a first order logic clause and I have to transform it to its normal clausular form. $$\forall x \exists y \left[A(x) \land \lnot B(x) \implies C(x,y) \land\exists zD(z)\right]$$ But I have ...
3
votes
1answer
51 views

Are these two equivalences really as “immediate” as Jean-Yves Girard claims?

Section 2.1.1 Phase spaces from Jean-Yves Girard's "Linear Logic : Its Syntax and Semantics" says A phase space is a pair $(M,\bot)$, where $M$ is a commutative monoid (usually written ...
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votes
2answers
54 views

Prove or disprove, Equivalence vs Implication? [closed]

Prove or disprove, for any universal set U and predicates P and Q [ ∃x∈U, P(x) ∧ Q(x) ] ⇒ [ ∃x∈U, P(x) ∧ (∃x∈U, Q(x)) ]
2
votes
2answers
114 views

Proving the roots of a polynomial are irrational

This is a homework question so I'm just looking for some guidance. Basically we are asked to write a step by step proof in the form of assume/then statements for: $\forall x \in \mathbb{R}, ax^2 + ...
3
votes
2answers
55 views

Confused about the use of variables w/ logical quantifiers

Sorry if this is a really dumb question, but... After reading How to Prove it, I've become a little confused. On page 70, an example stating something similar to this is provided: $[\exists x P(x) ...
1
vote
1answer
46 views

law of implication

I'm trying to follow the solution of an exercise that asks to use rules of inference to show that something is true but I don't know how to go from step 2 to step 3: Step 1 $\quad \forall x((\lnot ...
0
votes
1answer
33 views

$[\exists x \in U, P(x)] \implies [\forall x \in U, P(x)]$

I know this statement is not always true, but I'm having a hard time proving it. I'm also wondering what the difference between: $[\exists x \in U, P(x)] \implies [\forall x \in U, P(x)]$ and ...
0
votes
1answer
24 views

Mathematical Predicate logic

Let Graph(x) be a predicate which denotes that x is a graph. Let Connected(x) be a predicate which denotes that x is connected. Which of the following first order logic sentences DOES NOT represent ...
3
votes
3answers
72 views

Is there any false case for that: $\exists x \in D, \forall y \in D, P(x, y) \implies P(y, x)$?

Is there any false case for that: $\exists x \in D, \forall y \in D, P(x, y) \implies P(y, x)$?\ I just can get the true case.\ How can we define D and P in a false case?\ Thx guys.
1
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2answers
60 views

why$(\forall x \in U, P(x)) \implies (\exists x \in U, P(x))$ is false?

may I have a complete proof of that "$(\forall x \in U, P(x)) \implies (\exists x \in U, P(x))$ " is false? thx guys
1
vote
1answer
42 views

Why this is true?$\exists x \in U, [P(x) \land Q(x)] \Leftrightarrow [(\exists x \in U, P(x)) \land (\exists x \in U, Q(x))]$ [duplicate]

$$ [\exists x \in U, P(x) \land Q(x)] \Leftrightarrow [(\exists x \in U, P(x)) \land (\exists x \in U, Q(x))] $$
2
votes
2answers
43 views

Translating from english to a symbolic sentence?

How would I translate the following from english to a symbolic sentence with quantifiers. The universe of discussion is all real numbers. Every integer is greater than some integer. I did the ...
1
vote
1answer
71 views

Fitch-Style Proof Help

I'm having some trouble solving a Fitch Proof, Here's how far I've gotten. Any Help is appreciated. Thank You
0
votes
1answer
27 views

Constructing a model for formula in propositional logic

Construct a model for fomula $F = \forall x \exists y(p(x,y) \land \lnot q(y,x)) \land \forall x \exists y \lnot p(x,y)$. So if i want to construct a model i need my formula $F$ to be true in all ...
3
votes
1answer
108 views

Natural Deduction $(∀x∃y (P(x) → Q(y))) → (∃y∀x(P(x) → Q(y)))$ [duplicate]

I am having trouble with this Natural Deduction question $$(∀x∃y (P(x) → Q(y))) → (∃y∀x(P(x) → Q(y)))$$
1
vote
1answer
24 views

predicate logic ,writing in notation form

The statement below should be rewritten in the form “ for all · · · x, · · · .” "No computer scientists are unemployed" Answer Let computer scientists = CS unemployed=U for all x element of CS, x ...
1
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1answer
41 views

More questions on quantifiers

I have the following questions: Write the following statements in more abbreviated form, using quantifiers. Here the short phrases “is prime” and “is a line” are allowed, and the symbol $\Pi$ may be ...
0
votes
1answer
31 views

Meaning of this- $\forall x \in S, \exists y \in S, p(x,y) \implies \exists y \in S, p(y,y)$

The problem was to prove or disprove the following $ S\neq \varnothing,\, \forall x \in S, \exists y \in S, p(x,y) \implies \exists y \in S, p(y,y)$
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1answer
20 views

Ambiguous Sentence to Translate to Predicate Logic

I have been given the following sentence to translate into predicate logic: "Everyone can go home only if the work is done." I have come up with two interpretations: A) "Everyone can go home only ...
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2answers
68 views

Predicate logic, transitivity (sort of?)

I have a question. It involves 2 pictures for which I'm supposed to write a formula which is true for one, but false for the other. The pictures can be found here on page 23 (the arrows pointing in a ...