# Tag Info

### 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 ...
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### 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....
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### 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 ...
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### 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 ...
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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 $A$, ...
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### 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 ...
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### 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 ...
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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 $L(x,y)\... • 199k 2 votes Accepted ### On the statement$ \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 $$\exists \alpha>0 \quad \forall s>0 \quad \exists x>1 \quad\left|\int_1^... • 38.8k 2 votes Accepted ### What's the domain of x in the RHS of the following logical equivalence, \forall x \in D, P(x) \equiv \forall x, x \in D \rightarrow P(x)? As Mauro states in the Comments, it's all about interpretations. Take a logic statement like \forall x \ P(x) What does this say? We read this as 'All objects have property P". OK, but what ... • 99.4k 2 votes Accepted ### What is the error in this deduction of \exists x (P) \leftrightarrow P? Usually, in Hilbert-style proof systems, there are some restrictions regarding the interaction of Gen rule and Deduction Theorem. If Gen allows us to prove Γ,P⊢∀xPx without restrictions, we cannot ... • 93.9k 2 votes Accepted ### Problem understanding quantifier for x in epsilon-delta definition of a limit You can only quantify an assertion about 𝑓(𝑥) when 𝑥 is in the domain of 𝑓. The assertion you claim is not vacuously true for other values of 𝑥, it is nonsense. The function whose domain ... • 93.7k 2 votes Accepted ### Problem with converting predicate expression to Prenex Normal Form Observe that your result$$∃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∨... • 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$...
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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 ...