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Questions tagged [quantifiers]

The quantifiers $\forall$ ("for all") and $\exists$ ("there exists") distinguish predicate calculus from propositional logic.

Quantifiers specify the quantity of objects that satisfy a given formula.

The quantifiers $\forall$ (for all) and $\exists$ (there exists) are the most common, but others such as $\exists!$ (there exists a unique) are also in usage.

Only use this tag if your question is about the usage of a quantifier in a formula. Be sure not to use this tag for any question with quantifiers.

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Writing propositions using connectives and quantifiers

So I have this statement that says: For every two real numbers $x$ and $y$ with $x < y$, there is a real number with the property $P$ between $x$ and $y$. Let $P(x)$ be the statement that says that a real number $x$ has some property $P$. I have to…
Shaun Tolley
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Can the two place "Most C are D" quantifier be defined from the one place "For most X" quantifier?

The two place quantifier "All C are D" (where C and D represent classes) can be defined from the single place quantifier "For all X", like so: "For all X, if X is C, then X is D". Also, the two place quantifier "Some C are D" can be defined from the…
user107952
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What are the quantifiers in this statement?

Statement: Suppose $V$ is finite-dimensional with $\dim V \ge 2$. Prove that there exist $S,T \in L(V, V)$ such that $ST \ne TS$ I am confused about the first part "Suppose $V$ is finite-dimensional with $\dim V \ge 2$". Would the quantifier be…
Seeker
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Is "distributing" a quantifier logically equivalent?

Suppose I have this statement, $$∀a:∀b:P(a)→Q(b)$$ and then I have this statement $$(∀a:P(a))→(∀b:Q(b))$$ Are these two statements the same? While I'm not sure about a proof, nor am I sure about my explanation, but I feel like intuitively you can…
redrobinyum
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Existential quantifier question $\forall x \forall y(P(x)\land P(y) \implies x=y)$

Quesiton about existential quantifier The unique quantifier is $!\exists xP(x)$ so only one x is true. And I am wondering if the following is equal to unique quantifier $\forall x \forall y(P(x)\land P(y)) \implies x=y)$ So this is saying that for…
Fernando Martinez
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Which quantifier to use for the domain of tangent in set-builder notation?

I am trying to write out the domain of the function $\tan x$ in set-builder notation. I have seen this written out as $$\left\{x \in \mathbb{R} \mid (\exists n \in \mathbb{Z})\left[x \neq \frac{\pi}{2} + \pi n\right]\right\},$$ with the existential…
ericw31415
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Quantifiers solution writing

For a problem such as:$$\forall k\in\mathbb{N},\exists j\in \mathbb{Z},k=j+1.$$I can immediately see that this is false as it does not hold for all $k$, for instance when $j=-10$, $k$ becomes negative and that is not a natural number. However how do…
Aurora Borealis
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Empty set under existential quantifier

Let $U = \{s_1, s_2, s_\cdots, s_n\}$ be the universal set and $A = \{a_1, a_2, a_3,\cdots, a_m\}$ is the set under consideration. Now, I am proving $\forall_{x \in \{\}} p(x) = T$ as follows $$\forall_{x \in A} p(x) = \forall_{x \in u} (x \in A…
hanugm
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Explanation about quantifier sequence ∀x∃y and ∃y∀x

I'm learning the science of programming. And there are some quantifier questions I cannot understand, does anyone could explain? Let $p(u,v) \ ≡ \ v≤u^2 $ Is $(∀x.∃y.p(x,y))$ valid? Is $(∃y.∀x.p(x,y))$ valid?
Junrui
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Negation of a statement - use quantifiers

Let $Y \subseteq \mathbb{R}^n$. I have the following statement: "$Y$ satisfies constant returns to scale if $y \in Y$ implies $\forall \alpha \ge 0$ it follows that $\alpha y \in Y$." I am trying to find the definition of $Y$ does not satisfy…
elbarto
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Quantifiers and Truth Values

Determine the truth value of each statement, assuming that x and y are real numbers, and justify your answer. $\forall$x, $\exists$y such that xy=1 $\exists$y such that $\forall$x , xy=1 I understand the first problem to be true. Since the…
ErinA
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Can I use Existential Geneneralization on Universal sentences?

I have some doubts which I divided into two parts. Suppose the following universal sentence ∀x (Px → Qx) (i) Could I use universal generalization on it? I mean ∀x (Px → Qx) Px → Qx from 1, Universal instantiation …
Lauro Morais
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Why use "\exists x" in set-builder notation?

Why is the \exists k used in this set-builder notation?
waterlemon
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How to write Fermat's Last Theorem using quantifiers?

How do I write Fermat's Last Theorem using only universal and existential quantifiers. I only want to quantify it over the natural numbers (excluding zero)- for which I use $ℤ+$ to represent this set. My attempt at it, which I don't think is right,…
R.McGinnis
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