0
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1answer
31 views

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 ...
5
votes
2answers
66 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
40 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
votes
3answers
38 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 ...
1
vote
1answer
66 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 ...
0
votes
2answers
63 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
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 ...
2
votes
1answer
55 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
3answers
71 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
1answer
17 views

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 ...
-2
votes
4answers
170 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 $ [$\forall x \in \mathbb{R}$] ...
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 ...
1
vote
1answer
136 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
109 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
93 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? ...
5
votes
2answers
86 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 ...
4
votes
1answer
81 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 ...
1
vote
2answers
60 views

Discrete math question - nested quantifiers

question regarding nested quantifiers. $$\forall x \forall y\big((x < y)\to (x^2 < y^2)\big)$$ Determine the truth value for this question. I think this is false because if $x$ is $4$ and $y$ ...
0
votes
0answers
21 views

Nested quantifiers - Please help - Step by step [duplicate]

I'm fairly new to to this and have asked a question before on here but still don't understand how to determine the truth values of a nested quantifier. So for example $\exists y \forall x (x^2 < y ...
0
votes
1answer
98 views

Nested Quantifiers - true or false

I was just wondering what the truth value is for: $\exists y \forall x (x^2 < y + 1)$. The domain of discourse is: $R$ X $R$. The reason I believe this is false is x = y = 0. Which makes the ...
1
vote
1answer
72 views

Expressing statements using quantifiers [duplicate]

I would like to confirm my answers for these questions before carrying out with more studying--since there are no answers, and I'm worried I will be practicing incorrectly. Let F(x)="x is friendly", ...
1
vote
3answers
129 views

Converting statements with quantifiers

I'm having a little trouble understanding quantifiers and therefore doubting all my study answers. Since there is no where to check if the answers are correct, I'm very very worried I am just ...
1
vote
1answer
304 views

Determining truth value of quantified statements

Ok, sorry! I know I asked a question not but 1 hour ago, but I have one final question remaining about determining the truth value of a statement. I would again like confirmation of my answer for a ...
2
votes
2answers
297 views

Express the statements using quantifiers example

I'm having a little trouble understanding quantifiers and therefore doubting all my homework answers. Since there is no where to check if the answers are correct, I'm very very worried I am just ...
4
votes
1answer
138 views

Determining the truth value of certain quantifiers based on this proposition being false.

Can you help me verify if I answered this question correctly? Consider $[(\forall x)(P(x)) \land (\exists x)(\lnot Q(x))] \implies \{(\forall x)(P(x)) \iff [\lnot(\forall x)(R(x)) \lor ...
0
votes
2answers
85 views

How to take more than two in logical quantifiers

Let the universe of discourse be all humans. Let $F(x,y)$ denote $x$ is a friend of $y$. Stating the following logically: No one has more than two friends. $$ \neg ( \exists x \exists y \exists ...
1
vote
2answers
28 views

Does the Quantifier apply to all?

I have the following question: Let $f(x, y, z) = x^2y+z^3$, where $x, y, z \in \mathbb{Z}$. For each of the following determine its truth value. Justify your answers. (a)$\exists x, y, z: ...
5
votes
3answers
182 views

How can one simplify $¬(¬∃x, P(x)) $ and $\neg(\neg\forall x,P(x))$?

What I've learned so far: $\lnot$($\forall$$x$, P($x$)) $=$ $\exists$$x$, $\lnot$P($x$) $\lnot$($\exists$$x$, P($x$)) $=$ $\forall$$x$, $\lnot$P($x$) So far so good (I hope!) But what about ...
3
votes
3answers
56 views

Representing the statement using Quantifiers

I want to represent the statement "Some numbers are not real " using quantifiers. I have been told by my teacher that the correct way to represent this is num(x) : x is a number real(x) : x is real ...
3
votes
2answers
240 views

$\forall x \in I , \exists y \in I$ such that $xy \in I $

I just have a small question! Really basic I'm sure but something is bothering me. Take note of the following statement: $\forall x \in I , \exists y \in I$ such that $xy \in I $ Does this ...
3
votes
2answers
227 views

Translation of : The disjunction of two contingencies can be a tautology.

The statement is: "The disjunction of two contingencies can be a tautology." The predicates are: $C(x)$: "$x$ is a contradiction." $T(x)$: "$x$ is a tautology." The book says the answer is ...
1
vote
1answer
350 views

The truth value of quantified statements

I just took an exam and the following problems were asked: Determine the truth value of each of these statements if the domain consists of all real numbers. $\forall x \forall y \; ...
2
votes
1answer
389 views

Determining The Truth Value Of Quantified Statements

The problem I am working on is: Determine the truth value of each of these statements if the domain consists of all integers. a) $∀n(n+1>n)$ b) $∃n(2n=3n)$ c) $∃n(n=−n)$ ...
3
votes
1answer
1k views

How to use the Rules of Inference to a statement from two premises

The problem is as follows: Given the premise ∀x(P (x) ∨ Q(x)) and ∀x((¬P (x) ∧ Q(x)) → R(x)) is true, use the rules of inference to show that ∀x(¬R(x) → P(x)) is also true. (The domains of all ...
3
votes
4answers
412 views

Quantified Statements To English

The problem I am working on is: Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people. a) $∀x(C(x)→F(x))$ ...
5
votes
6answers
629 views

Quantifiers, predicates, logical equivalence

I am asked if $(\exists x) (P(x) \rightarrow Q(x))$ and $\forall x P(x) \rightarrow \exists xQ(x)$ are logically equivalent. I don't think they are but how will I prove it? Am I supposed to use ...
2
votes
2answers
208 views

Converting $\exists x \exists y (x\geq y)$ into English

$\exists x \exists y (x\geq y)$ The universe of discourse is all real numbers. This says that there exists an $x$ and there exists a $y$ such that $x\geq y$. But what is this actually trying to say? ...
3
votes
1answer
317 views

Are these statements logically equivalent? (quantifiers)

Is $\forall x(P(x) \vee Q(y))$ the same as $(\forall x P(x)) \vee Q(y)$? I understand that if I had $\forall x(P(x) \vee Q(x))$, that it is not the same as $(\forall x P(x)) \vee (\forall x Q(x))$. ...
1
vote
1answer
648 views

nested quantifiers

In the domain of integers, $P(x,y)$. predicate "$xy = 12$" I'm not sure why $(\forall x)(\exists y)P(x,y)$ is false statement. "For all $x$, there are some $y$, such that $xy = 12$". ex.: ...
2
votes
1answer
117 views

Why is it true that $∀x((-x)^2=x^2)$?

I'm trying to learn discrete math and I'm lost as to why this truth value is true. Can anyone please explain why? The domain consists of all real numbers. $∀x((-x)^2=x^2)$ The answer is True, but I ...
1
vote
2answers
115 views

Express each of these sentences in terms of $Q(x, y)$, quantifiers, and logical connectives,

Let Q(x,y) be the statement “x has been a contestant on quiz show y”, where the domain of x is the set of students and the domain for y consists of all quiz shows. For each of the English sentences ...
2
votes
3answers
137 views

Are these two statement equivalent?

$\forall x \exists y P(x,y)$ $\exists x \forall y P(x,y)$ where P(x,y) means x is smaller than y. I believe that they mean the same thing.
1
vote
3answers
180 views

Are these statements equivalent (quantifiers)?

$\neg \forall x \exists y \neg P(x,y)$ is equal to $\exists x \exists y \neg P(x,y)$ I had to make sure, because I wasn't sure at all.
0
votes
2answers
119 views

Is this equivalence true?

Is this equivalence true? $(\forall x (P(x)) \wedge (\exists y Q(y)) \equiv \forall x \exists y(P(x) \wedge \exists x Q(y))$ Here is what I did so far. If the LHS is true, then there exists a x ...
2
votes
3answers
105 views

What's the negative statement of

$\forall x: P(x) \rightarrow Q(x) \vee R(x) $ Is it for some $x$, if not $P(x)$, then not $Q(x)$ and not $R(x)$?
1
vote
2answers
207 views

Is double quantifying a variable possible in predicate logic?

I read it as "Everyone is either a student or has read every book". But what's the use of the existential y outside the bracket? ...
2
votes
3answers
162 views

Are these two predicate statements equivalent or not?

$\exists x \forall y P(x,y) \equiv \forall y \exists x P(x,y)$ I was told they were not, but I don't see how it can be true.
1
vote
1answer
136 views

Can a domain with universal quantification also be considered to have existential quantification?

If the predicate is true for all the domain (universal quantification), could it also be considered true for some of the domain (existential quantification)? Similar to a subset within a set? Or ...
1
vote
1answer
702 views