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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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Is the order of universal/existential quantifiers important?

If you have a formula with existential quantifiers, it is important in which order they appear. Just to make an easy example: $\forall$ man $\exists$ woman: the woman is the true love of the man which is obviously a different statement…
Martin Thoma
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7
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Why negating universal quantifier gives existential quantifier?

Negating a universal quantifier gives the existential quantifier, and vice versa: $\neg \forall x = \exists x \neg \\ \neg \exists x = \forall x \neg $ Why is this, and is there a proof for it (is it even possible to prove it, or is it just an…
George Newton
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How to read $(\forall e) ((\exists N=N(e)) (P(N,e)))$

Is there any difference between $(\forall e) ((\exists N=N(e)) (P(N,e)))$. and $(\forall e) ((\exists N)(P(N,e)))$ Should we read this two statements differently? What does this $N=N(e)$ stand for? To me it is clear that $N$ depends somehow on…
Evgenii.Balai
  • 299
4
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3 answers

Does universal quantification order matter?

For example if I have $$\forall a \in A \forall b_2 \in B \forall b_1 \in B((a,b_1)\notin R \lor (b_1, b_2)\notin R)$$ is is the same as $$\forall b_2 \in B \forall b_1 \in B \forall a \in A ((a,b_1)\notin R \lor (b_1, b_2)\notin R)$$ and other…
pseudomarvin
  • 861
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2 answers

Converting English to Quantifiers: 'There is no greatest prime'

I'm working on an exercise that appears rather simple, but the answer I keep coming up with differs from what the instructor found. Say I want to convert the sentence 'there is no greatest prime' to quantifier notation, and I'm to work with two…
user465188
2
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1 answer

Negate the following sentence: $\forall x, |x−a| < δ \Rightarrow |f(x)−L| < \varepsilon$

Negate the sentence: $\forall x, |x−a| < δ \Rightarrow |f(x)−L| < \varepsilon$ For my negation I got: $\exists x, |x-a| < δ \Rightarrow |f(x)-L| \geq \varepsilon$ Would that be correct?
Jake Park
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2
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1 answer

Quantifiers for multiple variables?

I know about the universal quantifier(translated to "for all") $\forall$ and the existential quantifier(there exists) $\exists$. But I am not sure what the correct way is to use them for multiple multiple variables. For example: How would you use…
That Guy
  • 1,309
2
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1 answer

Proof method for expression involving quantifiers

Assume that I want to prove $\forall x \forall y P(x,y)$ where $P(x,y)$ is some proposition. But, instead, if it were easier to prove $\forall y \forall x P(x,y)$ and if I prove the latter one just beacause of its easiness, will I also be proved…
hamsi
  • 23
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1 answer

Write with quantifiers P and ¬P

Let P: "for all natural $n$, exists an integer $p$ such that $np>2n $ Is…
B. David
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1 answer

What is the Negation of following statement

I faced a question in exam where we had to tick the correct negation of the following statement $"\exists y\in \Bbb{Z}, \forall x\in \Bbb{R}, \text{such that}\ y^2
blabla
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Nested quantifiers order?

$$a) \quad \forall x\exists y(3x+4y=12)$$ $$b) \quad \exists x\forall y(3x+4y=12)$$ Am I correct in saying that a) is correct while b) is false?
Heelloppp
  • 65
1
vote
1 answer

How do you negate the following sentence?

$ ∀x ∈ R,∃n ∈ Z ,x^{n}>0 $ How do you negate it so that the ¬ symbol does not arise to the left of any quantifier? Is the negated statement is true?
sydg
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1 answer

What does this quantifier evaluates to?

I need help solving the following question, Suppose U=Z. Simplify the quantifer, and say that if it is true or false with the help of and example, ~$\exists$x $(x| x|5 \Rightarrow x|15)$ Note that |= modulo. Now for all values it evaluates to…
user2857
1
vote
1 answer

Simple proof with quantifiers. Help me understand.

I have to negate $$ \forall x \in \mathbb{Z} \space \exists y \in \mathbb{Z} \space (( x \ge y) \land (x + y = 0)) $$ and prove either the original proposition or negation is true. I get the negation $$ \exists x \in \mathbb{Z} \space \forall y…
T L
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A clarification regarding the Existential Introduction rule

I require clarification on the Existential rule E4 in Eliot Mendelson's Introduction to Mathematical Logic, page 61: Let $t$ be a term that is free for $x$ in a wf $A(x,t)$, and let $A(t,t)$ arise from $A(x,t)$ by replacing all free occurrences of…
Katie Henebery
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