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

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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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1answer
29 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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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 ...
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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.
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How to interpret this index variable and corresponding summation limits?

Let $TD$ be an integer (That is, $TD \in \{ 1,2,3,4,\dots \}$). I am looking at the following statement: If $TD$ is even, the (solution) is given by $$ \forall \frac{TD}{2}\in \mathbb{N}: x_i^* = 1 - ...
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Simple proof involving quantifiers

Task: Prove this theorem: $ \exists x (P(x) \Rightarrow \forall yP(y) )$. I got this far: I figured out this is equivalent to $\exists x (\neg P(x) \lor \forall yP(y) )$. I don't understand ...
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How do I translate 'no philosopher student admires any rotten lecturer' into quantificational logic formula?

Let's assume that $Fx=x$ is a philosophy student, $Rx=x$ is a rotten lecturer, and $Mxy=x$ admires $y$. My translation of the sentence was $\forall x(Fx\supset\neg\forall y(Ry\supset Mxy))$, but my ...
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B is an element of some power set of A such that A is an element of F.

$$B \in \{\,\mathcal P(A) \mid A \in F\,\}$$ I can't quite figure this out. My textbook says this statement is equivalent to $$\exists A \in F\ \forall x\ (x \in B \leftrightarrow \forall y\ (y \in ...
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correct typesetting for quantifiers

For years I have been typing and writing quantifiers in a certain way. Now that I am writing my thesis, my adviser is taking issue with some of these things. Since he is my adviser I'm going to do ...
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How can I interpret a multiply-quantified statement?

∃ x ∈ R such that ∀ y ∈ R, x + y = 0. Can anyone help me rewrite this statement in plain english without symbols or variables? So far I have "There exists a real number whose number and other number ...
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“If everyone in front of you is bald, then you're bald.” Does this logically mean that the first person is bald?

Suppose we have a line of people that starts with person #1 and goes for a (finite or infinite) number of people behind him/her, and this property holds for every person in the line: If everyone ...
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Translating this nested quantifier to english (negation of nested quantifiers)

So i'm a total newb at this so I need help on one the questions for my assignment: Let S(x) = “x is a student at Bronx Community College”; F(x) = “x is a faculty member at Bronx Community College”, ...
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Universal Quantification of a Predicate P(x)

Consider the following statement: Every number is less than its square. Write the statement “Every number is less that its square” symbolically by defining a predicate and using a quantifier. Answer: ...
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How should I prove $\forall x \in \mathbb{Q}$ where $x > \sqrt 2$ , $\exists y \in \mathbb{Q}$ where $\sqrt{2} < y < x$

How should I prove the below statement? $\forall x \in \mathbb{Q}$ where $x > \sqrt 2$ , $\exists y \in > \mathbb{Q}$ where $\sqrt{2} < y < x$ I tried to prove it by contradiction ...
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Universal Quantifiers [duplicate]

What is the difference between ∀x∈U : ~p(x) and ~∀x∈U : p(x) ?? Could anybody give any English sentences explaining both of ...
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Did I correctly translate 'Whoever loves Myfanwy, loves a philosopher only if the latter loves Myfanwy too.' into quantificational logic?

There is an exercise in my textbook. Suppose ‘$m$’ denotes Myfanwy, ‘$n$’ denotes Ninian, ‘$o$’ denotes Olwen, ‘$Fx$’ means x is a philosopher, ‘$Gx$’ means x speaks Welsh, ‘$Lxy$’ means $x$ loves ...
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sets with quantifiers

My professor wrote the following theorems and definitions on the blackboard. ...
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Algebraically transform logic expression

Algebraically transform: $\neg \forall x(P(x) \wedge Q(y) \implies \exists zR(z))$ to $\exists x\forall z(P(x) \wedge Q(y) \wedge \neg R(z))$ Justify each step with one or more ...
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Determine whether or not $∀x[p(x) → q(x)]$ and $[∀xp(x)] → [∀xq(x)]$ are logically equivalent.

Determine whether or not $∀x[p(x) → q(x)]$ and $[∀xp(x)] → [∀xq(x)]$ are logically equivalent. I believe that they are not equivalent, but that is just an assumption. I am not sure how to go ...
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Write expressions w/out quantifiers (convert to AND/OR expressions)

A universe contains the three individuals $a,b$, and $c$. For these individuals, a predicate $Q(x,y)$ is defined, and its truth values are given by the following table: \begin{array}{c|ccc} ...
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Why is this predicate false?

I am stumped at my professor's answer to this predicate logic. all x and y are natural numbers. ∃y∃x(x >= y) I think it is true, since there is a pair ...
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Universal Instantiation and Proof of: if $x$ and $y$ are odd integers, then $xy$ is odd.

I have a question about the first part of the proof for the statement "If $x$ and $y$ are odd integers, then $xy$ is odd" if we are using a direct proof. Now i've gotten the proof roughly correct but ...
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finding a DNF with an expression that contains quantifiers

I am supposed to use equivalencies to find the prenex DNF for the wff: $\exists xp(x) \land \exists xq(x) \rightarrow \exists x(p(x) \land q(x))$ It's been awhile since I've done something like this ...
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Changing (enlarging) the domain in a Quantified statement

I would like to ask the following. If we have the proposition $$\forall x\in\mathbb{R}^{+}(x^2>0)$$ and we wish to use as a domain the $$\mathbb{R}$$ instead. Is it correct that it will translate ...
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Nested Quantification of exactly one.

Suppose my domain is "All students in the class" and P(x, y):= x has emailed y. So, how do i define: Every student has emailed exactly one student. Exactly one student has emailed every one. A ...
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Symmetric and transitive relation definition problem

Mathematical definitions and notation really confuse me. For example, a definition similar to the following can be found in many textbooks and online: In mathematics, a binary relation R over a ...
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Translating Mathematical (Universal/Existential Quantifier) Statements into English?

Can anyone confirm if I have these correct? Or if not, where I am going wrong? Translate these statements into English, where $K(x)$ is '$x$ is a Kangaroo' and $H(x)$ is '$x$ hops' and the domain ...
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Quantifier notation: $\forall n \implies \cdot$ versus $\forall n, \cdot$

I'm not sure which of the following two notations is the correct one (or, are both correct?). I've seen both being used by different professors. $\forall \varepsilon > 0\ \exists \bar n \colon ...
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Is the establishment of the validity of this argument correct?

I am trying to show that the following argument is valid. There is an email that is sent but it is not saved in the inbox. All emails are saved in the inbox or the inbox is full. If the inbox is ...
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nested quantifiers (exactly one questions)

Express this statement using quantifiers, without using the uniqueness quantifier."There is exactly one student in this class who has taken exactly one mathematics class at this school" T (x, ...
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Statements involving quantifiers

I am confused regarding the following; If we have a statement, for example, $$\exists_{x} \in X, \forall_{y} \in Y, x + y = 0.$$ Now, I'm wondering if you could just choose $x$ as $-y$, or do you ...
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Shifting bounded quantifiers

The universe of the following variables are the natural numbers $\mathbb{N}$. I found in the literature the following logic equivalence: $\forall n < k \exists m \ \varphi(m,n) \leftrightarrow ...
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Restricting universal quantifiers with conditions?

I want to express "For every $x\in\mathbb R$ that has (at least) one $p\in\mathbb Z$ and $q\in\mathbb N$, such that $x=\frac pq$, $x\in\mathbb Q$ is true" with logical quantifiers, just like this ...