25 votes

Does leaving universal quantifiers up to context lead to ambiguity?

Yes, it does generate ambiguity. Let $E$ and $O$ be predicates for "is even" and "is odd," respectively, and consider the difference between $$\forall x(E(x)\rightarrow O(x))$$ and ...
Noah Schweber's user avatar
12 votes
Accepted

Example of a quantifier which is not compatible with ordered pairs.

How about $\exists!$, the unique existence quantifier? Working in the structure $(\mathbb{N};<)$, we have that $\exists !x\exists !y(x>y)$ is true but $\exists !(x,y)(x>y)$ is obviously false....
Noah Schweber's user avatar
10 votes
Accepted

Difference between “for some $k$” and “for some arbitrary $k$”

In your first example, I would just write “where $k$ is an integer”. But “some” is okay. The point is, since $n$ is already known, $k$ is completely determined. Writing “where $k$ is some arbitrary ...
Michael Weiss's user avatar
8 votes
Accepted

Equivalent statements not giving equivalent negations

The negation of 'All of these are such and so' is 'Some of these are not such and so' So note: the 'of these' part is still the same. Of course! Quantifiers range over some domain. If not all objects ...
Bram28's user avatar
  • 99.4k
7 votes
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What is the difference between "for all" and "there exists" in set builder notation?

Neither are particularly accurate, but that's because in part I think you want $x \in \mathbb{Z}$ instead. I would also insist on the use of "such that" in lieu of "and" in $A$, ...
PrincessEev's user avatar
  • 43.4k
6 votes

Difference between "for any" and "for all"?

Here's a comparison of the various translations of $$∀x{∈}F\; P(x).$$ “For all elements $x$ in $F,\,P(x)$ holds” sometimes sounds like the property $P$ might belong to $F$ as a whole rather than to ...
ryang's user avatar
  • 38.8k
5 votes

Equivalent statements not giving equivalent negations

Since your "$\forall x:x>0\;P(x)$" (equivalent to the more usual $\forall x>0\;P(x)$) means $$\forall x\left(x>0\implies P(x)\right),$$ its negation is $$\exists x\left(x>0\land\...
Anne Bauval's user avatar
4 votes
Accepted

Is it valid to quantify two variables over the same universe using the same quantifier (like in $\forall a,b \in \mathbb{R} \:\: P(a,b)$)?

These are equivalent to one another $\forall a\,\forall b \; \big(a\in \mathbb{R} \land b\in \mathbb{R}\implies P(a,b)\big)$ $\forall a\,\forall b \; \big((a,b)\in \mathbb{R}^2\implies P(a,b)\big)$ $\...
ryang's user avatar
  • 38.8k
3 votes
Accepted

How to prove vacuous quantifier

If $\forall x Fx$, then any element works as a witness to prove the desired statement. On the other hand, if $\lnot\forall x Fx$, then there is an element $m$ such that $\lnot Fm$, and this element ...
Karl's user avatar
  • 11.3k
3 votes

Does leaving universal quantifiers up to context lead to ambiguity?

$ \forall x \forall y [\forall z (z \in x \Leftrightarrow z \in y) \Rightarrow \forall w (x \in w \Leftrightarrow y \in w)]\tag1$ This statement means $$ ∀w∀x∀y\color\red{\exists z}\;\Big( (z \in x \...
ryang's user avatar
  • 38.8k
3 votes
Accepted

Set of natural numbers as the intersection of inductive sets

Yes, $\forall n \in S(n^+ \in S)$ and $\forall n ((n \in S) \rightarrow (n^+ \in S))$ are equivalent (and usually the first is just short way to write the second). Property you use ($0 \in S \wedge \...
mihaild's user avatar
  • 15.3k
3 votes

What is the difference between "for all" and "there exists" in set builder notation?

Actually, $A=\left\{2,4,8,16,\ldots\right\}$, whereas $B=\emptyset$. This second assertion comes from the fact that there is no $n\in\Bbb N$ which is equal to every power of $2$ with natural exponent. ...
José Carlos Santos's user avatar
3 votes

What is the difference between "for all" and "there exists" in set builder notation?

When you're having trouble understanding the basic meaning of some expression like $$A = \{n \in \mathbb{N} : \exists x \in \mathbb{N} \text{ and } n=2^x\}$$ it can be helpful to separate the basic ...
3 votes
Accepted

"Some students in this class grew up in the same town as at least two other students in this class.”

Let $D=\{ x: x$ is a person$\}$ be the domain consisting of all people. We also define the following predicate symbols... $ \begin{array}{11} Txy: & \text{$x$ grew up in the same town as $y$} \\ ...
RyRy the Fly Guy's user avatar
3 votes
Accepted

Prove the formula $(\forall x Px \wedge \forall x Qx) \leftrightarrow \forall x [Px \wedge Qx]$

Below I offer a syntactic proof using a first-order logic natural deduction system and the following inference rules: $ \begin{array}{l} \text{- conditional proof (CP)} \\ \text{- universal ...
RyRy the Fly Guy's user avatar
3 votes

Using and Rather than Implies

In English, the first is read as "for all $z$, $z$ being in $x$ implies that $z$ is in $y$". This means that everything in $x$ is also in $y$. This second is read as "for all $z$, $z$ ...
Tbw's user avatar
  • 995
3 votes
Accepted

$\exists!x\exists!yP(x,y)$ is not the same as $\exists!y\exists!xP(x,y)$?

$\exists!x\exists yP(x,y)\land\exists!y\exists xP(x,y)$
bof's user avatar
  • 77.8k
3 votes

$\exists!x\exists!yP(x,y)$ is not the same as $\exists!y\exists!xP(x,y)$?

While bof's: $\exists ! x \exists y P(x,y) \land \exists ! y \exists x (P(x,y)$ is certainly very efficient, I would suggest: $\exists x \exists y \forall x' \forall y' (P(x',y') \leftrightarrow (x' = ...
Bram28's user avatar
  • 99.4k
3 votes
Accepted

Is "there exists at least $0$" the tautological quantifier?

See this Wikipedia page for a rather brief discussion of what are called counting quantifiers. I will follow its notation and write, e.g., $\exists^{\ge2}x\phi(x)$ as the formalisation of "there ...
Rob Arthan's user avatar
  • 48.1k
2 votes

There is some integer $n$ such that if $n > 2,$ then $n^2 = 2n.$

Expanding on Robert's observation: This is the given sentence, which is true: there is some integer $n$ such that if $n > 2,$ then $n^2 = 2n$ $\exists n{\in}\mathbb Z \:\big(n > 2\:\to\: n^2 = ...
ryang's user avatar
  • 38.8k
2 votes
Accepted

$x^2=4 \implies \exists x{=}2(x^2=4)$

$x^2=4 \implies \exists x{=}2(x^2=4)\tag2$ Statement $(2)$ is an (unusual) abbreviation of $$x^2=4 \implies \exists x \;(x=2\land x^2=4).$$ Putting $x=2$ shows that its consequent is (mathematically) ...
ryang's user avatar
  • 38.8k
2 votes

Equivalent statements not giving equivalent negations

Actually the “equivalent”**statement you made does not make any sense(it is not well formed), really it should be: $\forall \epsilon( \epsilon>0 \implies \exists …)$ negating this we get $\exists \...
Vivaan Daga's user avatar
  • 5,501
2 votes
Accepted

What does $\exists x\left(L\left(x,x\right)\wedge \forall z\left(L\left(x,z\right)\rightarrow \left(z=x\right)\right)\right)$ mean?

Yes, they are equivalent. In your first formula, we can move the $\forall z$ outside the parenthesis. And in the context of $\exists x\forall z$, we have that $L(x,x)$ is equivalent to $L(x,y)\...
Arthur's user avatar
  • 199k
2 votes
Accepted

On the statement $ \exists \alpha>0 \quad \forall s>0 \quad \exists x>1 \quad\left|\int_1^x \frac{f(t)}{(t-1)^\alpha} d t\right|<s$ and its negation.

Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. Write down the negation of the statement $$\exists \alpha>0 \quad \forall s>0 \quad \exists x>1 \quad\left|\int_1^...
ryang's user avatar
  • 38.8k
2 votes
Accepted

What's the domain of $x$ in the RHS of the following logical equivalence, $\forall x \in D, P(x) \equiv \forall x, x \in D \rightarrow P(x)$?

As Mauro states in the Comments, it's all about interpretations. Take a logic statement like $\forall x \ P(x)$ What does this say? We read this as 'All objects have property $P$". OK, but what ...
Bram28's user avatar
  • 99.4k
2 votes
Accepted

What is the error in this deduction of $ \exists x (P) \leftrightarrow P$?

Usually, in Hilbert-style proof systems, there are some restrictions regarding the interaction of Gen rule and Deduction Theorem. If Gen allows us to prove $Γ,P⊢∀xPx$ without restrictions, we cannot ...
Mauro ALLEGRANZA's user avatar
2 votes
Accepted

Problem understanding quantifier for x in epsilon-delta definition of a limit

You can only quantify an assertion about $𝑓(𝑥)$ when $𝑥$ is in the domain of $𝑓$. The assertion you claim is not vacuously true for other values of $𝑥$, it is nonsense. The function whose domain ...
Ethan Bolker's user avatar
  • 93.7k
2 votes
Accepted

Problem with converting predicate expression to Prenex Normal Form

Observe that your result $$∃x∀y∃z\,\big(¬Px∨(¬Qz∧¬Ry)\big)$$ and your workbook's result $$∃x∀y\,\big(¬Px∨(¬Qx∧¬Ry)\big)$$ are both equivalent to $$∃a¬Pa∨(∃a¬Qa∧∀a¬Ra),$$ which is equivalent to $$∃a¬Pa∨...
ryang's user avatar
  • 38.8k
2 votes
Accepted

What is the calculation law in proposition logical?

Check it intuitively : when we say $[\forall x P(x)]$ , we mean that $P(x)$ is true for all elements. When we negate it , we are saying that there is some element for which $P(x)$ is not true , hence $...
Prem's user avatar
  • 8,897
2 votes

What is the difference between "for all" and "there exists" in set builder notation?

The abbreviation $$∃x{∈}\mathbb N$$ is short for $$\text{there exists some natural $x$ such that$\ldots$}\\\text{for some natural $x,\ldots.$}$$ Since it isn't a sentence (and doesn't mean “some ...
ryang's user avatar
  • 38.8k

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