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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.

1826 questions
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Confusion between open and closed intervals

I'm trying to solve the following exercise: In each case below, say whether the given statement is true for the universe (0, 1) = {x ∈ R | 0 < x < 1}, and say whether it is true for the universe [0, 1] = {x ∈ R | 0 ≤ x ≤ 1}. For each of the four…
J.s
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(∃!x)A(x) is equivalent to (∃x)A(x) ∧ (∀y)(∀z) (A(y) ∧ A(z) ⇒ y = z)

I have to prove this using real world examples and I don't know how. I tried to do this with nicotine, but the unique existential quantifier confuses me.
Luzmaria Diaz
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Is there a predicate A(x,y) such that the statement ∀x ∃y: A(x,y) is true, while the statement ∃y ∀x: A(x,y) is false?

I don't even know where to start, my professor just kinda went so fast through this and didn't explain this. I know that it says "For all of x and some of y, but after that, I just get lost. What is A(x,y)? If you can please explain, I truly want to…
Luzmaria Diaz
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$\exists p \in A$, $\forall q \in A $ , $q\leq p$

If $A \subseteq \mathbb{R} $ $$ \exists p \in A, \forall q \in A , q \leq p $$ Can I just use a specific value for $p$ and arbritary value for $q$ to disprove this? $p = 3$ and $q = p + 1$, hence $q > p$ Also, how would should one go about this…
u123435
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Quantifiers for all x,y in R

Can I write $$\forall{ x, y \in{\mathbb{R}}}$$ Instead of $$\forall{ x \in{\mathbb{R}}} \, \forall{ y \in{\mathbb{R}}}$$ Since it is much shorter and a similar notation is usually used in sets?
Hans
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existential quantifier distribution over equivalence

As in the title: does the existential quantifier distribute over equivalence? Is this true: $\exists_{x} \left( \phi \left( x \right) \Leftrightarrow \psi \left( x \right) \right) \Leftrightarrow \left( \exists_{x} \phi \left( x \right)…
Piotr Kłosowski
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Nested Quantifiers variable order

I have some fundamental questions about nested quantifiers: $$\begin{array}{c|ccc}P&1&2&3\\\hline1&T&F&T\\2&T&F&T\\3&T&T&F\end{array}$$ Is $\exists x \forall y\ P(x, y)$ the same as $\exists x \forall y\ P(y, x)$? What determines the order in which…
Duxa
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Negation of a statement?

L = {M, w| M is a TM, w is a string, and for some nonempty strings x, y, w = xy, M accepts xyx and M rejects yxy}.. Im having issues representing this in mathematical form. For example what would the negation of "M accepts xyx" be? or "for some…
user126885
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Are these statements true or false? The universe of discourse is the set of all people, and T(x, y) means “x and y are twins.”

I'm in math proof and problem solving and would like someone to tell me if I am on track with these answers. The question is: Are these statements true or false? The universe of discourse is the set of all people, and T(x, y) means “x and y are…
S.E.G.
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Presenting well a sentence with quantifiers

What is the syntax rule to present a syntax with quantifiers ? Should we rather write : \begin{equation} \forall x\in \mathbb N\quad \exists y\in\mathbb N \quad x
E. Joseph
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$\forall x[P(x) \vee Q(x)]$ is not $\equiv (\forall xP(x) \vee ∀xQ(x))$

I have a question from a discrete math text, Determine whether $\forall x[P(x) \vee Q(x)]$ and $\forall xP(x) \to \forall xQ(x)$ have the same truth value. Thus far using the definitions from my book $\forall x[P(x) \to Q(x)] ≡ \forall x[\neg P(x)…
javaderek
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Adding and subtracting variables which are part of an interval

I am following a paper proof which starts with the following constraint: $$\forall v \in [a,b], \forall \tilde{v} \in [a,b]$$ $$f_{i}(v) \geq f_{i}(\tilde{v}) + (v-\tilde{v})g_{i}(\tilde{v})$$ In the proof, the author writes let $h = \tilde{v} - v$…
A K
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Difference between two statements/quantifiers?

1) $\forall \varepsilon > 0$, if $a + \varepsilon > b$, then $a > b$. 2) If $\forall \varepsilon > 0, a + \varepsilon > b$, then $a > b$. When I read these two statements, they seem to imply the exact same thing, but apparently they mean two…
mr eyeglasses
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proofs involving power sets and universal quantifiers

Im having trouble solving with a proof problem "A is not equal to the Null class then the intersection of class A is a set" and help on proofing this?
brett
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Difference between usages of ∀

Is (∀n ∈ R, A(n)) ∨ (∀n ∈ R, B(n)) the same as ∀n ∈ R (A(n)) ∨ B(n)). My belief is that it should be the same no?
jojanqo
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