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

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How to formulate this logic formula

The problem setting is very simple. Suppose we have three variables x, y and z and a constraints C/3 predicate that is satisfied by the three variables C(x,y,z), but C/3 might not be the only ...
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Express $\forall n \in \mathbb N$, $\exists m \in \mathbb N$, $n^4 = m^2$ in words without using the symbol $\mathbb N$

Express $\forall n \in \mathbb N$, $\exists m \in \mathbb N$, $n^4 = m^2$ in words without using the symbol $\mathbb N$. My Solution: For all $n$ that is an element of Natural number there is ...
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express in words without using the symbol N

Express $$\forall\ n\in\mathbb N\ \exists\ m \in\mathbb N: \ n^4 = m^2$$ in words without using the symbol $\mathbb N$.
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Is ∃xP(x) ∨ ∃xQ(x) the same as ∃xP(x) ∨ ∃yQ(y)?

Very simple question: is ∃xP(x) ∨ ∃xQ(x)the same as ∃xP(x) ∨ ∃yQ(y) Thank you.
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Logic question is it true

Exercise: $$\begin{align} (\forall x>2) &~:~ |x|<3 \tag{P1}\\ (\forall x\in\mathbb{R})(\exists\varepsilon > 0) &~:~-\varepsilon <x<\varepsilon \tag{P2}\\ (\forall ...
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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?
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How to approach conversions of statements using predicates, quantifiers, and logical connectives.

I have an example problem where I must use predicates, quantifiers, and logical connectives to convert the statements. The statement is... ...
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Prove $\exists_x(\phi(x) \rightarrow \psi) \iff (\forall_x\phi(x) \rightarrow \psi)$ using natural deduction

I want to prove $\exists_x(\phi(x) \rightarrow \psi) \iff (\forall_x\phi(x) \rightarrow \psi)$ where $x \notin FV(\psi)$ using natural deduction method. I was able to prove implication from left to ...
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How can I represent these discrete math statements with predicates and quantifiers?

Use the predicate $A(x, y)$ that denotes ("$x$ loves $y$") Someone loves a person who loves everyone. Everyone loves someone who is loved by someone. I know I need to use nested universal and ...
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Logical Notations for Descriptive Mathematical Statements

I'm studying Discrete mathematics and I'm faced with a problem of converting a descriptive mathematical statements into logical notation. Any help would be appreciated. Thank you. a). Any integer ...
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quantification domain of set theory formulas

Let ZFC set theory, what is the domain of quantification of a formula like $\forall x\phi(x)$? If the domain is the whole Von Neumann Hierarchy $V$ why it is not a problem that it doesn't form a set?
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Analyzing Set of all Perfect squares

I have been reading Velleman's How to Prove book and the following statement in the book confuses me: For example, suppose we wanted to define S to be the set of ...
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Interpreting logical forms involving quantifiers

I have been trying to translate these two logical form into English statements without using any quantifier laws: (a) ∃x∀y ¬L(x,y) (b) ¬(∃x∀y L(x,y)) where ...
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Expressing statements in positive way

I have been working on this problem from Velleman's How to prove book: Negate these statements and then reexpress the results as equivalent positive ...
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Why does the author define these “logical notations” for set logic with “if then” and &?

In Section 1.1 of "Set Theory for Computer Science", the author defines $ \forall x \in X. P(x) $ and $ \exists x \in X.P(x) $ as shorthand for $ \forall x.(x \in X \Rightarrow P(x)) $ and $ \exists ...
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Deciding between implication and conjunction

This is one of the solved problems in Velleman's How to prove book: Analyze the logical forms of the following statements: 1) John likes exactly one person. ...
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Analyzing the logical form of “All married couples fight”

This is one of the example problems in Velleman's How to Prove book: Analyze the logical forms of the following statements. All married couples have ...
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Concluding Truth Value from Universe of Discourse

I have been working on the following problem from Velleman's How to Prove book: Are these statements true or false? The universe of discourse is the set of ...
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Idiomatic mathematical english statement for ∃x[P(x) ∧ ∀y(P(y) → y ≤ x)]

I have been working on problems from Velleman's How to Prove book and hit upon the following problem: Translate the following statements into idiomatic ...
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Questions regarding Universal Quantifiers

The question is to show that: $$\exists x:(P(x) \implies Q(x))\qquad\equiv\qquad\forall x:P(x) \implies \exists x:Q(x)$$ First I use double negation to get to the universal quantifier since it ...
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Use propositional logic to solve

Argue that (∀x(P(x) ∨ ∃y P(y))) is equivalent to ∃x P(x) Can anyone please explain how to do it?
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Understanding logical form of “Nobody in the calculus class is smarter than everybody in the discrete math class”

I'm self studying How to Prove book and have been working out the following problem in which I have to analyze it to logical form: Nobody in the calculus class ...
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Analyzing Logical Forms involving quantifiers

I have been solving the following problem from How to Prove book: Analyze the logical forms of the following statement: Everyone likes Mary, except Mary ...
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Relation between existential and universal quantificator in category theory

Let $\mathscr C$ be a cartesian (i.e. with finite limits) category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$ (here $I$ denote the terminal object). Let $f:X\to Y$ and ...
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Translating to English: $\forall x ¬(\forall y P(x, y)) \rightarrow \forall x \forall y ¬P(x, y)$

$$\forall x ¬(\forall y P(x, y)) \rightarrow \forall x \forall y ¬P(x, y)$$ I'm trying to intuitively understand this idea by thinking about it in terms of English. The second half is easy. Where P ...
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Universal and Existential Quantifiers

Are there any examples of a predicate P(x ) of a variable x such that the truth value of P(x) remains invariant under exchange of the Universal Quantifier ∀ and the Existential Quantifier ∃ -thanks
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Axiom of Unions and its use of the existential quantifier

I'm reading Halmos's Naive Set Theory, and right now I'm on the section about the axiom of unions. As stated in the book, the axiom reads: For every collection of sets there exists a set that ...
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Definition and decidability of bounded quantifiers

Consider quantifier-free formulas $P(x,y) = Q(x,y)$ of Peano arithmetic. Consider $P(x,y),Q(x,y)$ to be terms composed of variables $x,y, \operatorname{succ}, +, \times$. Note that these are ...
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Can universal instantiation be used more than once?

I'm trying to follow a proof in a logic text and it seems like the author used universal instantiation twice to reach the needed result. I was under the impression that you could only use UI one time ...
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proof of quantifier elimination in theory of real closed field/reals and existential quantifier over atomic formula

Standard proof of quantifier elimination for theory of real closed field/reals uses induction, as in Wikipedia article (http://en.wikipedia.org/wiki/Quantifier_elimination#Basic_ideas). However, it ...
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Alternate translation for: “Every real number except zero has a multiplicative inverse.”

A given text states, “Every real number except zero has a multiplicative inverse" (where mul- tiplicative inverse of a real number x is a real number y such that xy = 1). It offers the following ...
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Existential introduction - required or not?

Consider a theory $T$ in first order logic, and a formula $C$. If there exist a proof of $C$, and that all formulas in $T$ and $C$, none of them contains $\exists$. The question is: does there always ...
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Use of prime symbol in proof writing

Questions: So I looked through my course notes and saw this proof. I understand the content, but I'm confused about the use of the prime symbol. If we say that there is some $j'$ such that it is a ...
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Quantifier question?

How would I do the following quantifier and their negation No one loves everybody. or could you say : everybody does not love someone? x is all people So in symbolic this would be $\forall x, ...
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Quantifiers bind tightly?

Is this true that it is a commonly agreed rule that $\forall x\in A:P(x) \wedge Q$ and $\forall x\in A:P(x) \Rightarrow Q$ should be interpreted correspondingly as $(\forall x\in A:P(x)) \wedge Q$ and ...
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More questions on quantifiers

I have the following questions: Write the following statements in more abbreviated form, using quantifiers. Here the short phrases “is prime” and “is a line” are allowed, and the symbol $\Pi$ may be ...
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Translating a sentence into a logical expression.

I am having trouble understanding the solution given for a problem in my discrete mathematics text book. Any help would be much appreciated. Question: Let L(x, y) be the statement "x loves y", where ...
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Semantics and Logical structure in Definitons

Continuation of Free and bound variables in "if" statements definitions: A number is even if it is divisible by $2$. The number is even if it is divisible by $2$. Is the usage of the ...
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Free and bound variables in “if” statements

The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion $\forall x(x>2)$ is false. In the first case ...
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Logical structure of definitions

Here are some "concepts" that are confusing me: The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion ...
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$\exists \implies \forall$

I want to see some example theorem, when existence implies universality, so $\exists \implies \forall$ is true. I think matematical induction is a related technique, but I just don't see that ...
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Prove $\neg\exists x \neg P(x)$ is a logical consequence of $\forall x P(x)$

How would you prove that 2. is a logical consequence of 1. using a Fitch style proof? $\forall x P(x)$ $\neg\exists x \neg P(x)$
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Proof of Drinker paradox [duplicate]

I searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...
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Sets and Quantifiers

I have difficulty converting sentences into 'mathematical language'. According to my work above ...
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Truth set of a Universal Quantifier and Family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A(A\in F \implies x\in A)\bigr\} $$ If my understanding serves me correctly this notation means that all the $x$ ...
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Definition of intersection of a family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A: (A\in F \implies x\in A)\bigr\} $$ If my understanding serves me correctly this notation means that all the $x$ ...
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Set Theory Elementhood Notation

From How to Prove it: Given $A=\{n^2|n \in N\}$ where $N$ is the set of all natural numbers. I want to express A in terms of elementhood test notation. Velleman says $A=\{x| \exists n \in N ...
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Difference between bound and free variable

What is the difference between $\forall x (P(x)\implies Q(x))$ and $P(x)\implies Q(x)$ I know in the first one the variable x is bound but in the second one the variable is free. What are the ...
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Quantifiers and negation range

Hello people I have a simple question. I have this formula from which I need to remove all the implications. Here it is. $\forall x ( [ Roman(x) \wedge know ( x, Marcus )] \rightarrow [ hate (x, ...
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Symbolic logic proof

Can any one please give me the correct proof for this, i got this far but i am stuck. Thank you!