1
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1answer
51 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
93 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
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2answers
46 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
49 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
46 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
20 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
39 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
90 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
44 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
65 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
32 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
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2answers
71 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
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3answers
62 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
69 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
66 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
63 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
93 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
102 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
103 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. ...
2
votes
1answer
86 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
37 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
87 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
297 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
117 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
64 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
3answers
56 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
73 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
33 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
77 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
127 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
89 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
80 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
64 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? ...
0
votes
1answer
188 views

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 ...
2
votes
3answers
131 views

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?
0
votes
1answer
29 views

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 ...
1
vote
1answer
62 views

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 ...
1
vote
1answer
65 views

Logic/Quantifiers and Proofs/counterexample

How do I negate the following statement? Also please help me with this exercise:
4
votes
2answers
81 views

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 ...
1
vote
2answers
39 views

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 ...
2
votes
1answer
37 views

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 ...
2
votes
2answers
120 views

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 ...
0
votes
1answer
45 views

Why does $(\forall y)({\sim}Exy)$ have no quantifier-free equivalent statement?

Given a language $L=\{E, =\}$ where $E$ is an equivalence relation. Why does the statement $(\forall y)({\sim}Exy)$ have no quantifier-free formulation? Isn't: $$({\sim}Exa \mathbin\& {\sim}Exb ...
0
votes
1answer
51 views

Is this symbolic expresson correct?

Hello. I'd like to check my answer for 1. (g) $\forall$ x $\in$ A, P(x), $\forall$ y $\in$ A, C(y) $\wedge$ F(y), $\forall$ z $\in$A, C(z), T(x, y) $\implies$ T(x, z) Is this correct? Thank you
3
votes
1answer
88 views

Proof with quantifiers

$(\forall x)(\exists y)(x+y=0)$ $x$ and $y$ are real numbers The statement reads: for all $x$ there exists some $y$ such that $x+y=0$ is true. My proof is: take $y=-x$ Is this valid? I'm just ...
0
votes
1answer
35 views

Is this negation correct for this statement?

¬(for all n in N, there exists m in N, g(m,n)) equivalent to: there exists n in N, for all m in N, ¬g(m,n) Is that correct? Thanks!
4
votes
1answer
77 views

Negating $(\forall a \in A)(\exists b \in B)(a \in C \leftrightarrow b\in C)$?

I'm not quite sure how to go about doing this. When negating I know the quantifiers themselves will be negated meaning that $\forall$ would become $\exists$ and vice-versa. Also I know that ...
0
votes
1answer
34 views

Define domain $X,$ predicate $A(X)$ and $B(X)$

I'm having trouble creating a domain $X$ and the predicates $A(X)$ and $B(X)$ to for this set of sentences to be evaluated to be true or false. $(T)\quad \forall x \in X, (A(x) \rightarrow B(x))$ ...
0
votes
4answers
276 views

Express using logic symbols:

I have to also decide whether it is true or not after I express it in logic symbols. This is what I have so far..Am I correct? and I don't know how to do a).. a) There is a smallest positive number ...
0
votes
1answer
182 views

Logical Symbol Statements True/False?

so I'm working through homework questions for proofs class and unsure if I'm correct in my interpretation. I would really appreciate feedback. The questions states: Write the full meaning in English ...