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

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General conceptual confusion relating to vacuous proofs and quantifier help

I need to prove the statement: Let $x \in \mathbb{R}$. Prove that $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. So I start with the forward implication: If $1 ≤ ...
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Finding Truth Values Of Nested Quantifiers

I'm looking at for example, $∃x∀y,P(x≥y+1)$ I'm told in order to prove that this is true I can us the technique that follows: Find one value of $x∈X$(only needs to be one) that has the property that ...
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How is a quantifier-free formula actually interpreted?

My understanding is that a quantifier-free formula in FOL is simply a formula that contains no quantifiers, just possibly free variables. How is such a formula interpreted? My understanding is that if ...
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Relational calculus / predicate logic: basic semantics [closed]

Say I have part of a query in the form: $∃x_a(...)∧∃x_b(...)∧∃x_c(...)$, where $a$, $b$, and $c$ are attributes and the ellipses can be anything (I'm looking for a general rule). Is this equivalent to ...
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What is the difference between these two propositions? [duplicate]

My text says: Let Evens be the set of even integers greater than 2, and let Primes be the set of primes. Then we can write Goldbach’s Conjecture in logic notation as follows: $ \forall n \in ...
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Stuck on a quantifier logic problem

I've been trying to prove this to no avail.. $\vdash\exists x(Px\rightarrow\forall xPx)$ The book gives a hint.. that it might be helpful to prove the following two before tackling the main problem: ...
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Explanation of ordering of quantifiers in Real Analysis

Why is the second part where the quantifiers are interchanged false? Is there a concrete example for this?
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Equivalent logical quantifier statements?

I was doing an exercise that said convert the statement "Jane saw a police officer, and Rodger saw one too" into the logical equivalent using quantifiers. My answer was: $$ \exists x(P(x)\implies ...
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Logical Quantifiers

I am wondering if there is a reference or book that clearly translates all English forms of logical quantifiers to mathematical quantifiers. For example, when we say for any element $ x \in S$, is ...
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Predicate logic describing a function that is not onto.

I'm trying to understand how to write predicate logic describing a function that is not onto. A function is onto if every element in the codomain gets mapped to by some element in the domain using ...
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How to disjunct $\forall x.(P(x) \lor Q(x)) $

I really don't understand how to disjunct this. The whole argument is: $$\forall x.[P(x) \lor Q(x)] \rightarrow \neg[\exists x.P(x)] \rightarrow \forall x. Q(x) $$ Am I supposed to use the ...
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Prove $\forall x~\forall y~\forall z (x+y)+z=x+(y+z), \forall x~\forall~y\exists z~ x=y+z, \forall x~\forall z \exists y x=y+z ⊢ ∃y∀x x+y=x$

I need help using the standard rules of predicate logic with quantifiers to prove $~\forall x~\forall y~\forall z ~~(x+y)+z=x+(y+z), ~\forall x~\forall y~\exists z ~~x=y+z, ~\forall x~\forall z~ ...
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Interpretation of quantifiers

While studying quantifiers I got all confused with the following explanation about the order of quantifiers. The statement ∀x ∃y, y > x claims that, for any real number x, there is a number y ...
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How do I use rules of inferences to imply a conclusion from 4 premises?

I am a little confused on how to use 4 premises to prove a conclusion. Can you please tell me if my logic is sound for the following proof: ...
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Evaluating the reception of (epsilon, delta) definitions

There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. ...
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Are the following statements correctly translated?

Using predicate symbols shown below and appropriate quantifiers, write each English language statement as a predicate wff. Domain is all the objects in world. B(x) : x is a bee F(x) : x is a ...
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negation a logical statement/sentence with quantifier without universe of discourse

For example, $(\exists x) \,\,\forall y \in Y \,\, P(x,y)$. Here $\exists x$ does not have universe of discourse . In this case, can normal rule for negating the sentence/statement still be used? ...
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Question about quantifier logic

This is my first post on the mathematics stack exchange so please bear with me.. I am new to quantifier logic and I just can't seem to wrap my head around it. I have been given four statements and I ...
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Using quantifiers to express this sentence.

These are from a study guide, just checking my work. Let $F(x,y)$ be the statement "$x$ and $y$ are friends." where the domains consists of all people in the class. Use quantifiers to express the ...
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Why negating universal quantifier gives existential quantifier?

Negating a universal quantifier gives the existential quantifier, and vice versa: $\neg \forall x = \exists x \\ \neg \exists x = \forall x$ Why is this, and is there a proof for it (is it even ...
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$\lnot \exists x (\forall y (\alpha)\land \forall z(\beta) )\;$ is logically equivalent to which one of these?

These are the options: $\forall x(\exists z(\lnot \beta)\rightarrow \forall y(\alpha))$ $\forall x(\forall z(\beta)\to \exists y(\lnot\alpha))$ $\forall x(\forall y(\alpha)\to \exists z(\lnot ...
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Get rid of an existential quantifier

I have to remove the existential quantifier from the following formula: $$\exists i\left[\left(i \geq 0\right) \land \left(z-2i = 0\right) \land \left(y+i=x\right)\right]$$ First I make some simple ...
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Proving formally

$((\exists x : X.P) \Rightarrow (\forall x: X.Q)) \vdash (\forall x: X. PvQ) \Rightarrow((\forall x: X.P) \vee(\forall x: X. Q)$ exist stands for the existential quantifier all stands for for-all ...
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Sets and quantifiers question

Am I doing this correctly? Let S be a non-empty set, and let P(x) and Q(x) be open sentences that can be applied to any x∈S. For each of the following implications, determine whether or not it is ...
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∀x ∀y Q(x; y) What is the meaning

What is the meaning of ∀x ∀y Q(x; y)? Does this mean that: For all values of X every value of Y will satisfy Q(x;y)? so if Q(x;y) = x + y = x * 2 in this case ∀x ∀y Q(x; y) would be false? ...
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What is the purpose of universal quantifier?

The universal generalization rule of predicate logic says that whenever formula M(x) is valid for its free variable x, we can ...
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Translations from English to predicate logic, using quantifiers.

Let "book" be the set of all books and "Author" be the set of all authors. $\in$ denotes set membership. Consider the following predicates: short(x) is a predicate indicating x is a short book. ...
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Skolemization of a Formula

I have the following formula: forall x(p(x) <- exists y q(y,x)) What is the Skolemization of the above formula?
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Meaning of duplicated predicate quantifiers

What is the meaning of duplicated predicate quantifiers? Examples: $$ ∃x\ ∃x\ ∀x\ ∀x\ P(xy) \\ ∀y\ ∀x\ ∃x\ ∀x\ ∃x\ ∃y\ ∃x\ P(xy) \\ ∃y\ ∀y\ ∀x\ ∃x\ ∃y\ ∀y\ ∀x\ ∃x\ ∃x\ P(xy) $$
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Can the material implication ever be used as the main connective within the scope of an existential quantifier?

Can the material implication ever be used as the main connective within the scope of an existential quantifier? Usually, a conjunction is the main connective in sentences bound by an existential ...
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What does this combination of symbols mean?

I just want to know what this combination of symbols means: ∃! I know ∃ means 'there exists', but what does it mean when it is paired with a '!'? I have written down 'there exists unique" but I am ...
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Are the following statements TRUE OR FALSE: [closed]

Are the following statements TRUE OR FALSE: [$\forall x \in \mathbb{R}$] [$x > 0$ $\implies $ $x^2 > x$] [$\forall x \in \mathbb{R}$] [$x > 0$] $\implies $ [$\forall x \in \mathbb{R}$] ...
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Write the negation using logic symbols.

1) $(\exists x \in R)[(x^2 = (x+1)^2 ∧ (x^3 \in Z))]$ ATTEMPT : $((∀ x \in R)[(x^2 \not= (x+1)^2 ∧ (x^3 \notin Z))])$ 2)$(∀x \in R)(x>0) ⇒ (\exists n \in N)(n . x >1)$ Note: the (n.x) is ...
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In predicate logic, is it possible to distribute quantifiers

Is possible to establish that $\forall x \,\exists y\,(Fx \rightarrow Gy)$ is logically equal to $\forall x\,Fx \rightarrow \exists y\,Gy$? If it does not work, why not?
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Write the negation:

Write a negation of the following statement without using words of negation: A bounded real function cannot be surjective." Which is true, the statement, or its negation? Justify your answer. ...
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Nested Quantifiers - Differentiating between $\forall x \forall y$, $\forall x \exists y$, and $\exists x \exists y$

I have a few questions regarding quantifiers which I'm still not clear about. 1) $\forall x \forall y (x^2 + y^2 = 9)$ I believe this is false as x and y could be 2 and results in 8. 2) $\forall x ...
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First order logic. Describe that a set has more than 2 elements.

I would like to describe that a set has at least 3 elements using first order logic, would this be a valid way to do that? $\forall x\exists y\exists z(\neg(x=y)\wedge\neg(x=z)\wedge\neg(y=z))$ I ...
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Writing statements into symbols Discrete Math

The variable $x$ represents stduents, $F(x)$ means "$x$ is a freshman", and $M(x)$ means "$x$ is a math major" a) some freshme are math majors? $\exists x:F(x) \implies M(x)$ b) Every math major is ...
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Write the negation of the following statement (in words):

"For any field $F$, and any $a\in F$, if $a^3 = 1$ then $a = 1$." Is this statement TRUE OR FALSE? Is the negation TRUE OR FALSE? Attempt: There is a field $F$ and there is an $a \in F$ such that ...
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Unable to understand combination of quantifiers and set notation

I know what universal and existential quantifiers are but following is confusing,may be its comibination of set notation and quantifers. What does the following statement means? ...
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Are these equivalent?

$\forall x \in D, (P(x) \Rightarrow Q(x))$ is equivalent to $(\forall x \in D \cap P,Q(x))$. However, is this also equivalent to $(\forall x\in D)( P(x)\land Q(x))$? If not, what's the difference? ...
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Using implication with the Universal quantifier

While reviewing my AI textbook, I came across a paragraph that baffled me. It attempts to explain why the truth table for implication turns out to be perfect, as ...
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Is the following expression a tautology?

$\forall x\,(P(x)\rightarrow Q(x))\rightarrow (\exists y\,P(y)\rightarrow\exists z\,Q(z))$ I believe the sentece is a tautoloogy. Can someone confirm?
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Notation for exists two elements in a set with properties

I'd like to say: for any x in set X, if x is colorful, there must be t1 and t2, both in set T, such that t1 < t2 and green(x,t1) and red(x,t2). I believe this is the correct notation, but I'm not ...
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In predicate logic, can existential variables be used interchangeably?

When doing a derivation in predicate logic, am I allowed to use two different existential variables interchangeably? For instance, is $\forall xPx$ (or $∃xPx$) equal to $\forall yPy(∃yPy)$? If I ...
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Logic/Quantifiers and Proofs/counterexample

How do I negate the following statement? Also please help me with this exercise:
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Discrete math logic question

I have the following two questions. For all real numbers x, there is a real number y such that $2x+y=7$ would this be true or false? I think true because if you put $2(7)+y=14$ $2(8)+y=14$ there ...
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Negating statements / Finding $(A \cap B)',A \oplus B$ if $A=\{x \in\Bbb R \mid -3\le x\le0\}$ and $B=\{x \in \Bbb R\mid -1 < x < 2\}$

I am a bit new on this field and I am trying to solve some questions. I don't really think they are hard but there are some key points that I don't get it or I am stuck. Lets see. 1) Write the ...
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Express lattice axioms using implication and universal quantification

I'd like to ask for some help with homework. My task is to express lattice axioms in signature $(\leq, =, \sup, \inf)$ using only implication and universal quantification. Here are these axioms in ...
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Second order logic and quantification over formulas

According to Wikipedia second order logic allows quantification over sets of individuals and thus goes beyond first-order logic, e.g. in expressive power. On the other hand some sort of ...