Movatterモバイル変換

Question 1

1vote

1answer

118views

Is there any difference between "all" and "every, here? [duplicate]

I know that "all" refers to a complete group and "every" refers to each member of a complete group. But I am asking about the difference between them in a math context.For ...

A12345

823

asked20 hours ago

Question 2

0votes

1answer

68views

Generalization rule for universal quantifier and extended language

Let L be a language of first order logic. The generalization rule for universal quantifier says that, for $\phi\in L$, $\Sigma \vdash_L \forall x\phi(x) $ iff $\Sigma\vdash_{L\cup c}\phi([c/x])$, with ...

MarcoM

1

askedNov 22 at 19:46

Question 3

4votes

2answers

144views

Uniqueness quantification example

I have encountered a quite simple sentence, which is:The polynomial $x^2-6x+9$ has exactly one root in $\mathbb{R}$.If I were to write this using quantifiers it would obviously be:$\exists!_{x \in \...

Adamat

57

askedOct 31 at 0:24

Question 4

21votes

1answer

1kviews

What is the formal and rigorous definition of a quantifier?

In mathematical logic, there is the notion of quantifiers. The most commonly used ones are $\forall$ and $\exists$, but there are others, like $\exists!$, the unique existential quantifier, and the ...

user107952

24.8k

askedOct 28 at 18:51

Question 5

1vote

1answer

84views

Quantifier distributivity dilemma?

From the distributive property of quantifiers, we know,$$\forall\ x\in\ X,\left(P\left(x\right)\land Q\left(x\right)\right)\equiv\left(\forall x\in X,P\left(x\right)\right)\land\left(\forall x\in X,Q\...

Dipanjan Das

61

askedOct 16 at 6:59

Question 6

0votes

1answer

129views

Justifying universal generalization after existential elimination using dependence between quantifiers

In an attempt to transform following equivalent definitions of the cartesian product from the first into the second, trying to be rather formal about it:$$\forall A : A \in X \times Y \...

Or Gold

23

askedOct 2 at 0:36

Question 7

1vote

1answer

136views

Translating "Someone has visited every country in the world except Libya"

Let $V(p, c)$ mean that person $p$ has visited country $c$ in the world.Is the following deconstruction correct?Someone has visited every country in the world except Libya.There is a person $p$ ...

Saaqib Mahmood

27.8k

askedSep 12 at 9:43

Question 8

1vote

2answers

133views

Translating "Some triangles are green" [duplicate]

I have a pretty basic but tricky question about predicate logic.$T(x) : x$ is a triangle$G(x) : x$ is greenUsing the above predicates, I am pretty sure that the translation of the sentence "...

Frederic Verduyn

27

askedAug 22 at 15:09

Question 9

1vote

2answers

348views

Vacuous truth in the definition of differentiability

I have some knowledge of what vacuously true means: almost every case can be simplified as$\forall x$ $P(x) \Rightarrow Q(x)$, if $P(x) = \bot$, then $P(x) \Rightarrow Q(x)$ is always true no matter ...

H Mong

586

askedAug 18 at 2:51

Question 10

10votes

1answer

460views

Is this correctly worded? "If there exist integers $m$ and $n$ such that $12m + 15n = 1$, then $m$ and $n$ are both positive."

In "A Transition to Advanced Mathematics", 8th edition, by Smith, Eggen and St. Andre, exercise 1(f) in chapter 1.6 on page 60 says:Prove that if there exist integers $m$ and $n$ such ...

Ben

758

askedJul 1 at 23:58

Question 11

1vote

1answer

84views

Topological Equivalence

I'm currently looking at when two topologies $\tau(\mathscr B)$ and $\tau(\mathscr B')$ generated by bases $\mathscr B$ and $\mathscr B'$ are identical, and for this, we have:$$\tau(\mathscr B) = \...

Hermi

1,107

askedJun 13 at 14:40

Question 12

2votes

3answers

302views

Understanding the scope of quantifiers

I’m working through Example 2.3 in a logic text 1, which presents the formula$$\forall x.\;p\bigl(f(x), x\bigr)\;\to\;\Bigl(\exists y.\;p\bigl(f\bigl(g(x,y)\bigr), g(x,y)\bigr)\;\land\;q\bigl(x, f(...

desert_ranger

405

askedJun 13 at 13:56

Question 13

-1votes

1answer

66views

A problem about formalization of an argument with First Order Predicate Logic [duplicate]

I have task regarding formalization in first order predicate logic.The natural language goes:"Peter Petson is either completely tone-deaf or is listening to Professor Charles. Everyone who ...

emma

35

askedJun 10 at 10:39

Question 14

3votes

1answer

179views

$ ∃x \:(Φ(x) ∨ Ψ(x))$ versus $∃x \:∃y\: (Φ(x) ∨ Ψ(y))$

I'm learning about moving quantifiers to the front of logic formulas (prenex normal form). I have this formula:$$∃x\, Φ(x) ∨ ∃x\, Ψ(x)$$My conversion attempts have produced two different results, ...

Astro Biz

61

askedJun 3 at 15:05

Question 15

2votes

2answers

451views

Do we need to follow the quantifier order when applying a theorem/lemma?

TL;DR: I understand that the order of the quantifiers is important in understanding mathematical statements, but it seems that we can totally ignore this order when applying mathematical statements? I ...

linear_combinatori_probabi

1,630

askedMay 29 at 17:36

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Stack Exchange Network

Questions tagged [quantifiers]

Is there any difference between "all" and "every, here? [duplicate]

Generalization rule for universal quantifier and extended language

Uniqueness quantification example

What is the formal and rigorous definition of a quantifier?

Quantifier distributivity dilemma?

Justifying universal generalization after existential elimination using dependence between quantifiers

Translating "Someone has visited every country in the world except Libya"

Translating "Some triangles are green" [duplicate]

Vacuous truth in the definition of differentiability

Is this correctly worded? "If there exist integers $m$ and $n$ such that $12m + 15n = 1$, then $m$ and $n$ are both positive."

Topological Equivalence

Understanding the scope of quantifiers

A problem about formalization of an argument with First Order Predicate Logic [duplicate]

$ ∃x \:(Φ(x) ∨ Ψ(x))$ versus $∃x \:∃y\: (Φ(x) ∨ Ψ(y))$

Do we need to follow the quantifier order when applying a theorem/lemma?

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