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0answers
10 views

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 ...
5
votes
2answers
63 views

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 ...
1
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1answer
37 views

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 ...
0
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3answers
37 views

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 ...
3
votes
1answer
76 views

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 ...
2
votes
2answers
44 views

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 ...
1
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1answer
41 views

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 ...
2
votes
4answers
103 views

$\exists \implies \forall$

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 ...
3
votes
1answer
61 views

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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4answers
418 views

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 ...
3
votes
3answers
62 views

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 $x$ ...
3
votes
1answer
71 views

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 $x$ ...
3
votes
5answers
86 views

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 ...
1
vote
1answer
96 views

Symbolic logic proof

Can any one please give me the correct proof for this, i got this far but i am stuck. Thank you!
3
votes
3answers
83 views

Domain of discourse and quantifying in predicate logic

I am struggling with an idea about how quantifiers relate to domains of discourse. Given a statement "$x$ is divisible by $2$" represented by the predicate $D(x)$,the predicate currently has no truth ...
1
vote
1answer
49 views

Is there way to classify the quantifier rank $m$ first order sentence in first order logic

In its simplest situation, for example, if the signature contains only a binary relation $\sigma$, so the signature $\tau = \{ \sigma \}$, what are the inequivalent classes of all first order ...
2
votes
2answers
53 views

Commutivity of unique existence quantifiers

Find an expression P(x,y) to disprove the following equivalence, $(\exists!x)(\exists!y)P(x,y)\Leftrightarrow(\exists!y)(\exists!x)P(x,y)$ I could only think of a few statements of $P(x,y)$ that ...
1
vote
1answer
64 views

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 ...
1
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3answers
106 views

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 ...
0
votes
2answers
48 views

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: ...
0
votes
2answers
56 views

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 ...
1
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2answers
58 views

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 ...
0
votes
1answer
23 views

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 ...
1
vote
1answer
52 views

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 ...
1
vote
1answer
107 views

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~ ...
2
votes
1answer
53 views

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: ...
1
vote
2answers
84 views

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 ...
2
votes
1answer
43 views

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? ...
1
vote
2answers
79 views

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 ...
1
vote
3answers
69 views

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 ...
0
votes
4answers
75 views

$\lnot \exists x (\forall y (\alpha)\land \forall z(\beta) )\;$ 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 ...
0
votes
4answers
85 views

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 ...
0
votes
2answers
69 views

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 ...
1
vote
1answer
117 views

∀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? ...
2
votes
2answers
111 views

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 ...
0
votes
2answers
142 views

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. ...
3
votes
2answers
103 views

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?
3
votes
2answers
41 views

Meaning of duplicated predicate quantifiers

What is the meaning of duplicated predicate quantifiers? Examples: $$ ∃x\ ∃x\ ∀x\ ∀x\ P(xy) \\ ∀y\ ∀x\ ∃x\ ∀x\ ∃x\ ∃y\ ∃x\ P(xy) \\ ∃y\ ∀y\ ∀x\ ∃x\ ∃y\ ∀y\ ∀x\ ∃x\ ∃x\ P(xy) $$
4
votes
2answers
97 views

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 ...
4
votes
3answers
368 views

What does this combination of symbols mean? $\exists !$

I just want to know what this combination of symbols means: $\exists !$ I know ∃ means 'there exists', but what does it mean when it is paired with a '!'? I have written down 'there exists unique" ...
-2
votes
4answers
161 views

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 $ [$\forall x \in \mathbb{R}$] ...
1
vote
1answer
75 views

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 ...
2
votes
5answers
84 views

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?
2
votes
1answer
81 views

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. ...
0
votes
1answer
41 views

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 ...
2
votes
3answers
78 views

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 ...
1
vote
1answer
135 views

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 ...
2
votes
1answer
107 views

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 ...
0
votes
2answers
90 views

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? ...
3
votes
1answer
67 views

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? ...