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

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Set theory: how to interpret multiple quantifiers

How does one interpret this ZFC Union axiom? I can't quite understand what is meant after "There exists some elements y for all elements z"? I'm also wondering if the x is a typo. $\exists y \...
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First Order Logic Double Implication [closed]

I have a Logic Assignment of First Order Logic that I have to prove an initial claim, but one of the equations is kind of confusing for me because it has double implication and quantifiers. $$\...
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Quantification = statement about an open sentence?

The book I'm reading is talking about quantification being a method to convert open sentences into statements. From what I can see this method boils down to making a statement about the solution set ...
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Discrete Mathematics - Quantifiers problem

This is a question from the Discrete Mathematics question from Kenneth Rosen book. I didn't understand the question and thus I am confused how to begin with question. Below is the question from the ...
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Quantifiers Kenneth Rosen Discrete Mathematics

Please help me in regard with this question.I didn't have a clue how to solve this. The way I thought about this question is assuming the truth values of predicates P(x) and Q(x) and then trying ...
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negating propositional formula with quantifiers

In order to solve an exercise in computer sciences (proving a language $L$ to not be context-free) I need to negate the Pumping-Lemma. I was provided with the definition in the following form: If $L$ ...
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I have a question involving logical quantifiers which I have been stuck on for a while. I have trouble understanding the concept.

There is a question that has been bothering me where the concept is confusing to me. Assume B is the set of all boys and G is the set of all girls. L(B,G) represents that B likes G. $$\forall b \in ...
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Universal and Existential quantifier in Propositional logic

The following paragraph is an excerpt from Discrete Mathematics book of Kenneth Rosen 7edition The restriction of a universal quantification is the same as the universal quantification of a ...
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What is the formal definition of repeated limit?

The basic question is what has been asked in the title. I looked for the definition here, here and here but no definition uses quantifiers. I tried to formulate the definition but succeeded only ...
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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 ...
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How can I translate this sentence into predicates and quantifiers?

sentence : Every cube is larger than something else. My Working: P(x) = x is larger than something else ∀xP(x) But the answer is something completely different. ∀x (A(x) → B(x)) : the answer ...
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Predicate Logic - Archimedes' Library

Problem: Our class takes a field-trip to Archimedes’ Library. Before entering the library, your tour guide makes you notice the sign on the main doors which reads: “Observe the Rule of Archimedes’ ...
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order of multiple quantifiers

Problem: For a, b, c, d restricted to the universe of positive integers, explain why ∀a ∃b ∀c ∃d a/b = c/d is true, but ∀a ∃d ∀c ∃b a/b = c/d is false. I understand that the order of ...
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Reasoning informally about $\{x \in B \mid x \notin C\} \in \mathscr P(A)$

Attempting to apply more flexible, informal reasoning to predicate logic as demonstrated helpfully to me by another user in answer to my last question. $\{x \in B \mid x \notin C\} \in \mathscr P(A)$ ...
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Rewriting $\mathscr P(\bigcup_{i \in I} A_i)\not\subset\bigcup_{i \in I} \mathscr P(A_i)$ in more fundamental terms.

Working through Velleman's "How to Prove It" and currently having a bit of difficulty with a problem where I'm asked to rewrite this: $$\mathscr P\left(\bigcup_{i\in I} A_i\right)\not\subset\bigcup_{...
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How can I express each of these quantifications in English?

Let T(x) be the statement "x has visited Tashkent" where the domain consists of all students of my school. How can I express each of these quantifications in English? ...
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What is the negation of ∀x∃y¬P(x,y) without using ¬?

Found it to be ∃x∀yP(x,y). Is this right?
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Negation of double universal quantifications

In logic, when I want to negate the formula $$\forall x \forall y( F(y) \land A(y) \to \neg G(x,y))$$ what is the correct equivalent? Intuitively, I think it gives $$\exists x \forall y (F(y) \land ...
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Is this conclusion via rules of inference correct?

Use rules of inference to show: ∀x(P(x) → Q(x)) premise ∀x(Q(x) → R(x)) premise ¬R(a) premise ¬P(a) conclusion I have a lot of trouble with these sort of questions and was wondering if I did this ...
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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<y\qquad (1) \end{equation} or \...
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Nested Quantifiers (And vs Implies)

I would like to understand something regarding the nested quantifiers in discrete math. In the following question part (c): Let $M(x,y)$ be "$x$ has sent $y$ an e-mail message", and $T(x,y)$ be "$...
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why is $\forall x (p(x) \implies q(x)) \not\equiv (\forall x p(x)) \implies (\forall x q(x))$

I'm having a hard time wrapping my head around why $\forall x (p(x) \implies q(x)) \not\equiv (\forall x p(x)) \implies (\forall x q(x))$
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What does a period in between quantifiers mean?

I'm currently reading the notes (rather a book) of an MIT preliminary math course for discrete mathematics. In section on page 39, some ZFC axioms are written and roughly explained. For example, the "...
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Compound Quantifier

Can any one help me what will be the universe discourse of these two statements? if both statement has natural numbers or same universe of discourse what will be values, that makes 1st statement true ...
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Decidability of quantifier-free formulae in Peano- and True Arithmetic

It is well-known that validity in Peano Arithmetic is undecidable. It is less well-known that validity is already undecidable in True Arithmetic (the theory of the standard model of Peano Arithmetic). ...
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Existential and Universal Quantifiers

Quantifiers (a) Please see below. I cannot work out why one is correct. If $x < 0$, then there's no value $y \in \mathbb{R}$ so that $y^2 = x$. (b) If I have $\exists$ followed by $\forall$, then ...
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Definition of rational number in logical expression format

So I have to translate the following definition of rational number into logical expression. The real number r is rational if there exist integers p and q with q = 0 such that r = p/q. I have ...
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How to express the following statement with Quantifiers and Predicates

Use quantifiers and predicates with more than one variable to express this statement: There is a student in this class who has taken every course offered by one of the departments in this school ...
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Is there any linear algebra textbook presented using logical symbols?

I'm currently going through a book called Linear Algebra Done Right by Axler, and to be honest, his book seems to be very loose with what things he defines. For instance , the symbol 0 could be mean a ...
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logic: order of quantifier with free variables

Take the sentence, "You can't win them all." This could be logically written as "For all people, there exists a thing they cannot win at." $\forall x.\exists y.(\neg win(x,y))$ Now suppose I was ...
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Negation of a quantified statement regarding an implication.

I am trying to improve my understanding of the negation of a quantified statement where the statement is an implication. I am doing a practice problem which I dont have the answer to from the textbook ...
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Question on negation of a quantified statement

There is a slight confusion I am having when comparing my answer to a solution for a problem. Basically, the question asks me to state the negation of For every integer $n$ such that if $n$ is ...
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Is the universal quantification symbol $\forall\;$ known as “for any x” or “for all x” in First Order Logic & why is this different to Discrete Math?

I'm reading a book by Mark Tarver called Logic, Proof and Computation. Chapter 8 (starting p71) is about First Order Logic. On page 77 (of Chapter 8) the author writes: For any value for 'x', in ...
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Quantifier elimination, $(\mathbb{R}, <)$

I have a general question about quantifier elimination. Which kinds of formulas do you have to observe? For example let T be the theory of $(\mathbb{R}, <)$ and I want to show, that this theory ...
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Converting ∃ to ∀ and vice versa

I'm having some trouble getting my head around the conversion of quantifiers. For instance, I know that $\forall x \,F \,\equiv\, \neg\exists x \, \neg F$ and conversely. $\exists x \,F \,\equiv\, \...
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Symmetry and transitivity with the existential quantifier

I can't find any resource that would indicate whether symmetry and transitivity relations are possible or not with the existential quantifier or when quantifiers are combined. I'm interested in the ...
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Quantifier in Set definitions

Can the definition be made more readable: $\overline{R_1} = \{ (j_1,j_2)\mid j_2 < j_1 + \Gamma^r_a + c^r_w \text{ and } j_1 \in J^r_v \text{ and } j_2 \in J^r_v \text{ and } a = (v,w) \in A^r ...
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Question about universal quantifier

when I was reading a paper about the universal quantifier, I met this equation, says we can do conversions like the following: A -> B ≡ ¬A v B can anyone help ...
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true or false statements based on predicate logic

Q) Let $P(x,y)$ be the predicate $y=2x$. Consider the statements a)$\forall x \exists y P(x,y)$ b)$\forall y \exists x P(x,y)$ c)$\exists y \forall x P(x,y)$ where $x$ and $y$ range over the ...
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Is this statement logically true? If so How?

Q) Is the statement (∃xQ(x)∧∃xR(x))↔∃x(Q(x)∧R(x)) logically true? If it is, explain why. If it is not give an interpretation under which it is false. I have asked this previously but did not get an ...
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translating uniqueness quantifier algorithmically

Given a claim with a uniqueness quantifier $\exists$!, such as: $$\forall x \exists!y \ P(x,y) $$ A standard translation that uses $\forall$ and $\exists$ only (there are several possible ones) is: ...
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Proving/disproving statements with a given context of natural numbers.

How do I prove the following statements or their negations in the context where $x$ and $y$ are rational numbers in the closed interval $[-\sqrt{2}, \sqrt{2}]$? Statement 1: $\forall x \exists y\; x &...
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Predicate logic: Symbolize a sentence using a dictionary and two-place predicates

Given the following dictionary, how would the sentences below be translated in to a language using quantifiers? My attempts are shown as well: Dictionary: $L$: a two place predicate which means -...
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Alternatives to pure quantifier logic

Are there some alternatives for pure quantifier logic? Pure quantifier logic is axioms and rules of inference added to proposition logic to result first order logic. Are there other axioms that ...
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Trouble understanding negation of definition of convergent sequence.

Definition of convergent sequence: $$\forall \varepsilon >0, \exists N \in \mathbb R: \forall n \in \mathbb N \ (n \ge N \implies d(x_n,x) < \varepsilon)$$ I found the negation to be: $$\exists ...
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Composing relation with identity yields original relation

Let $\phi: F\rightarrow G$ be a relation and $id_F$ be the identity relation on $F$. Then $\phi\circ id_F = \phi$ . Attempted proof: $$\phi\ \circ id_F = \{(f,g)\in F\times G: \exists f_2\in F((f_1,...
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Prove $\forall X [X \subseteq A \land y \in A] \rightarrow \exists X[X \subseteq A \land y \in X]$

Prove: $\forall X [X \subseteq A \land y \in A] \rightarrow \exists X[X \subseteq A \land y \in X]$ proof: Proving by contradiction, suppose $$\forall X [X \subseteq A \land y \in A] \text{ and } \...
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quantifiers that bind nothing

Consider $\forall x \forall y P(y,y)$, where $x$ is quantified, but does not appear. Is quantifying a variable that otherwise does not appear well-formed or meaningful? I've seen this sort of thing ...
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Discrete Mathematics - How to Express the phrase “There is no one who did action y”

Let's say $P(x,y)$ means $x$ sent an e-mail to $y$. If I want to say that no one has sent a message to Jean, then aren't there multiple ways to do this? $\neg Ǝx(P(x, Jean))$ But I can also say $∀...
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conditional proof with different variable [closed]

I don't know how to prove $(Ǝx~F(x)~→~Ǝx~G(x))$ with conditional proof from: $(Ǝx~F(x) ~→~ ∀z~H(z))$ $H(a)~→~G(b)$ I have a problem with different variables here and conditional proof as well.