# Tagged Questions

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

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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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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. $\exists x \,F \,\equiv\, \... 0answers 22 views ### Symmetry and transitivity with the existential quantifier I can't find any resource that would indicate whether symmetry and transitivity relations are possible or not with the existential quantifier or when quantifiers are combined. I'm interested in the ... 1answer 30 views ### Quantifier in Set definitions Can the definition be made more readable:$\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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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: ...
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: $\forall x \exists y\; x &... 2answers 25 views ### Predicate logic: Symbolize a sentence using a dictionary and two-place predicates Given the following dictionary, how would the sentences below be translated in to a language using quantifiers? My attempts are shown as well: Dictionary:$L$: a two place predicate which means -... 1answer 73 views ### 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 ... 2answers 26 views ### 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: $$\exists ... 0answers 19 views ### Composing relation with identity yields original relation Let \phi: F\rightarrow G be a relation and id_F be the identity relation on F. Then \phi\circ id_F = \phi . Attempted proof:$$\phi\ \circ id_F = \{(f,g)\in F\times G: \exists f_2\in F((f_1,... 2answers 27 views ### 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 } \... 1answer 39 views ### 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 ... 1answer 31 views ### 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$∀...
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.