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Questions tagged [logic-translation]

For translating between natural language expressions and logic expressions.

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Negating "He will sink unless he swims" [closed]

I want to negate "He will sink unless he swims" using the formula $$\neg (P \Rightarrow Q) \equiv P \wedge (\neg Q).$$ But first, how do we write that statement as if-then statement?
Babai's user avatar
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1 vote
4 answers
157 views

Symbolising an following argument with two Therefore's

I am trying to translate the following argument to logic symbols to verify its validity using truth tables: If the supplier supplies the seeds, then if the seeds are sown on time, then the plants ...
Trascendence's user avatar
1 vote
1 answer
39 views

Writing the definition of Upper Bound

Let X be an ordered set. Let $ S \subset X.$ An element $ u \in X$ is said to be an upper bound for $S$ if $s \leq u$ for all $ s \in S.$ In first-order logic, how do I write the above definition? Is ...
Dr. J's user avatar
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3 answers
161 views

Choosing between the 'and' and 'implies' connectives

$pol(x): x$ is a politician $liar(x): x $ is a liar All politicians are liars : $\forall x(pol(x) → liar(x))$ Some politicians are liars : $\exists x(pol(x) \land liar(x))$ No politicians are liars :...
user1327299's user avatar
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1 answer
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Do these two logic transaltions have the same meaning?

From Rosen's Discrete Math textbook: Translate the statement “Every real number except zero has a multiplicative inverse.” (A multiplicative inverse of a real number $x$ is a real number $y$ such ...
Bob Marley's user avatar
2 votes
1 answer
38 views

Maximizing Perimeter of Triangle PDE on a Parabola and Finding Coordinates of N for Rhombus Formation

Given a parabola in the Cartesian plane defined by the equation ( $y = -\frac{1}{2}x^2 + \frac{3}{2}x + 2 $), it intersects the x-axis at points A and B, and the y-axis at point C. Consider a point P ...
Oth S's user avatar
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1 answer
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Counting the numbers of quantifiers, how are there 4?

From the book "Gentle Introduction to Art of Mathematics": How many quantifiers does this have and what kind? "Everybody has some friend that thinks they know everything about a sport.”...
Yanjan. Kaf.'s user avatar
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2 answers
259 views

Do ∃x(Dog(x)) and ∃x(¬Dog(x)) contradict each other? [closed]

Formally, ∃x(Dog(x)) and ∃x(¬Dog(x)) look like they contradict each other. However, in the real world, there exist objects which are dogs i.e. ∃x(Dog(x)) there exist objects which are not dogs i.e. ∃...
Boris Rusev's user avatar
5 votes
4 answers
226 views

On the tautology $(P \implies Q) \vee (Q \implies P)$

The logical statement $$(P \implies Q) \vee (Q \implies P)$$ is an example of a tautology. However, if I choose logical statements for $P$ and $Q$, it is not always true that either $Q \implies P$ or $...
Lauren S's user avatar
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When to use implication arrow versus equivalence arrow?

In my class we've been asked to complete an exercise and choose whether to use implication or equivalence arrows: "The equation $2x−4=2$ is fulfilled only when $x=3$." I understand that we ...
kads's user avatar
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4 answers
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Finding the relationship (equivalence or implication) between two expressions

I am trying to find the relationship between $$\exists X \; (p(X) ∧ q(X))$$ and $$\exists X \; p(X) ∧ \forall X \; q(X).$$ I believe that quantifiers cannot be used in forming truth tables, after all ...
Govt_employee's user avatar
1 vote
1 answer
173 views

Two arguments with the same form, one valid, one not

A convertible car is fun to drive. Isaac’s car is not a convertible. Therefore, Isaac’s car is not fun to drive. Letting $C(x)$ be "$x$ is a convertible car" and $F(x)$ as "$x$ is fun ...
Akash Shanmugaraj's user avatar
9 votes
5 answers
3k views

Is "William only eats icecream when the sun is shining" a biimplication?

William only eats icecream when the sun is shining Let $P(t)$ be the sun is shining at time $t.$ Let $Q(t)$ be William is eating an icecream at time $t.$ Which implication is there between $P(t)$ and $...
Marcus K. Johnson's user avatar
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1 answer
259 views

Using quantifiers to rewrite some simple mathematical statements

Have I used quantifiers correctly to rewrite these sentences? The equation $x^3=7$ has at least one root. $∃x∈\mathbb R\;\;x^3-7=0$ The equation $x^2-2x-5=0$ has no rational roots. non$(∀x∈\mathbb ...
Marcus K. Johnson's user avatar
1 vote
1 answer
91 views

Translating "there is no number between two consecutive numbers"

There is no number number strictly between two consecutive number. Is my translation $$∀x ∈ Z | ∀y ∈ Z | ¬∃z ∈ Z \;(z<y ∧ z>x ∧ y=x+1)$$ correct? Is $$∃x | ¬∃n ∈ Z \;(n<x<n+1) $$ also ...
Cody Bijeaux's user avatar
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1 answer
55 views

Not knowing how to translate a theorem to logic [closed]

Proposition: for every positive integer $n$, there do not exist four positive integers $a,b,c,d$ with $ad=bc$ and $n^2 <a<b<c<d<(n+1)^2$ I understand how to prove this by looking for a ...
Si_monster's user avatar
11 votes
3 answers
1k views

Difference between “for some $k$” and “for some arbitrary $k$”

I am told that the “for some” and “for some arbitrary” are different. For example, when proving the statement “if n is odd, then $n^2$ is odd”, one of the steps includes writing $$\text{$n = 2k+1,\:\:...
bluesky's user avatar
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2 answers
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Symbolising the set of pupils who do not like both subjects

If $A= \{\text{pupils who like Science}\}$, $B= \{\text{pupils who like History}\},$ then is the set of pupils who do not like both subjects $$(A\cap B)^\complement$$ or $$(A\cup B)^\complement\,?$$
lay's user avatar
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1 answer
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What does the word 'with' mean in this theorem?

If $a, b$ and $c$ are positive integers with $a, b ≥ 2$, then equation (1.1) has at most one solution in positive integers $x$ and $y$ with $b^y ≥ 6000 c^{1/δ∗(a,b)}.$ I'm unclear about the above ...
Nimish's user avatar
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Translation of : "the sum of two prime numbers each greater than 3 is even" in First Order Logic

I have used these symboles : $Z(x) : x$ is a prime number $f(x,y) : x + y$ $L(x,y) : x\ge y$ $t: 3$ $E(x) : x$ is even $$∀x∀y(Z(x)∧Z(y)∧L(x,t)∧L(y,t))\rightarrow E(f(x,y)))$$ Is this a correct ...
Lzkb's user avatar
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1 answer
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Formalising the sentence "someone is plotting against me"

Someone is plotting against me. Can the above sentence be translated into predicate logic without using the existential quantifier? If not, is it because the sentence is self-referential?
Richard Gasquet's user avatar
3 votes
3 answers
619 views

Is there a symbol for "as long as" in math?

Let's say we have the expression $$∀𝑎,𝑏∈ℤ:𝑎𝑥=𝑏𝑥⟹𝑎=𝑏$$ Which means "for all values of 𝑎 and 𝑏 in the set of integers, if 𝑎𝑥 equals 𝑏𝑥 then 𝑎 must equal 𝑏." For example, if 4×...
The_Animator's user avatar
1 vote
3 answers
82 views

How to translate SAME and DIFFERENT in predicate logic?

Betty told every secretary a lie. I was told that this sentence is ambigous as it can be intrepreted in 2 different ways. These are my 2 interpretations: "Betty told every secretary the same ...
Cira's user avatar
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2 votes
2 answers
242 views

Translating "There are two different students in your class who between them have sent an e-mail message to or telephoned everyone else in the class"

Let $M(x,y)$ be “$x$ has sent $y$ an e-mail message” and $T(x,y)$ be “$x$ has telephoned $y$”, where the domain consists of all students in your class. Assume that all e-mail messages that were sent ...
Eric's user avatar
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To prove that for every $x$, $(x\in Z\implies x\in R),$ is it ok to write "For any $x,$ suppose $x\in Z$. Then... Then $x\in R$"?

To prove that for every $x$, $(x\in Z\implies x\in R),$ is it ok to write "For any $x,$ suppose $x\in Z$. Then... Then $x\in R$" ? For example, in the above proof of $$\forall x{,}y\left(\...
lightyourassonfire's user avatar
2 votes
1 answer
135 views

"There are two different people who have visited exactly the same websites"

Let $W(x,y)$ mean that student x has visited website y, where the domain for $x$ consists of all students in your school and the domain for $y$ consists of all websites. Express the statement $∃x∃y∀z((...
Eric's user avatar
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2 answers
194 views

Which of these translates the universal and existential quantifiers better?

Consider this proposition (which I know is false): $$(\exists y{\in}\mathbb Z)\,(\forall x{\in}\mathbb Z)\,(y > x).$$ I am wondering whether the analogy of picking a variable value according to the ...
Princess Mia's user avatar
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4 votes
2 answers
324 views

Translating "Boris hasn't tried anything better than chocolate"

Boris hasn't tried anything better than chocolate. The above sentence needs to be converted to first-order logic. The given domain is candies, and Boris is a sentient piece of candy. $B(x,y)$: $x$ is ...
MrExp's user avatar
  • 51
2 votes
2 answers
163 views

Formalising "every number other than $0$ has a unique multiplicative inverse"

The domain is the set of all real numbers. Express "Every number other than 0 has a unique multiplicative inverse" as a logical expression. My solution is: ...
work work's user avatar
2 votes
1 answer
36 views

Hows my symbolization for the following Argument?

Every philosophical empiricist admires Humes. Some philosophical idealists like no one who admires Humes. Therefore, some philosophical idealists like no philosophical empiricists. $\forall x(Ex \...
JCAL's user avatar
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1 vote
1 answer
58 views

I am Confused, mixing predicates and propositional letters?

In the exercises at the end of each section the author gives you the Arguments and ask you to determine the validity of it, he then goes on and gives you the proper predicates to use which are located ...
JCAL's user avatar
  • 244
1 vote
1 answer
107 views

Is my symbolization of the following argument correct?

Formalise the following argument, then prove or disprove its validity: Some psychologists admire Freud. Some psychologists like no one who admires Freud. Therefore some psychologists are not liked by ...
JCAL's user avatar
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1 vote
1 answer
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Translating "None of the paintings is valuable except the battle pieces."

Example 1: No intelligent person who drinks to excess also eats to excess. I am stuck on deciding whether this means a) $\forall x(Ix \implies -(Dx \lor Ex)$ or b) $\forall x(Ix \land Dx \implies -...
JCAL's user avatar
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1 vote
1 answer
72 views

Translating "If there were no computers with antivirus every computer would work fine."

"If there were no computers with antivirus every computer would work fine". Use: O(x) = x is a computer, A(x) = Computer x has an antivirus, F(x) = x works fine. My take is $$∀x((O(x)∧¬A(x))...
PMathC's user avatar
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0 votes
1 answer
79 views

Negation of converse of a theorem related to linear independence

Theorem $T:\quad$ Let $S = \{v_1, v_2,..., v_p\}$ be a set of vectors in $\mathbb R^m$. If $p>m$, then this set is linearly dependent. Theorem $T$ can be proved to be true. Converse $T_c$ of ...
Vinod's user avatar
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2 votes
1 answer
62 views

What does $\exists x\left(L\left(x,x\right)\wedge \forall z\left(L\left(x,z\right)\rightarrow \left(z=x\right)\right)\right)$ mean?

I translated "There is someone who loves no one besides himself or herself" as $$\exists x\left(L\left(x,x\right)\wedge \forall z\left(L\left(x,z\right)\rightarrow \left(z=x\right)\right)\...
Bezbiletnik's user avatar
0 votes
1 answer
114 views

"No integers $x$ and $y$ exist for $28x+7y=8$"

What is the logical structure of this statement? No integers $x$ and $y$ exist for $28x+7y=8.$ I'm not sure, but I think the answer is $$¬∃x\;∃y\;(x ∈ \mathbb Z ∧ y ∈ \mathbb Z ∧ 28x + 7y = 8).$$
user1161390's user avatar
5 votes
2 answers
175 views

How do I translate "is not sufficient" into symbolic logic?

Take the proposition "it is not sufficient for the monkey to dance in order for me to get an A on the test" m = the monkey dances a = I get an A on the test It makes sense why I can ...
Ben's user avatar
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1 vote
2 answers
86 views

"Neither Ana nor Bob can do every exercise but each can do some."

I’m a bit confused as to how I should translate the following sentence: Neither Ana nor Bob can do every exercise but each can do some. I've identified the atomic sentences $A$ = Ana can do every ...
Natalie Alexandra Anghel's user avatar
1 vote
1 answer
99 views

Expressing the instance of a ZF Set Theory axiom for a given property

I am currently in the process of studying ZF Set Theory (without the Axiom of Choice) and I have come across a type of question that is unclear to me. The basic format of the question is to "...
FD_bfa's user avatar
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3 votes
1 answer
108 views

Predicate Logic "John drinks any type of drink"

For the following English statement,what would be the correct predicate logic translation: John drinks any type of drink. 1.$\forall x {(Drinks(John,x))}$ 2.$\forall x(Drink(x) \to Drinks(John,x)\\$ I ...
Don Su's user avatar
  • 77
1 vote
2 answers
70 views

Have I misunderstood this argument about truth values? [closed]

7.2.1 Every truth value is confirmed by intuition that affirms it. $(∀x)(Tx\to Ix)$ 7.2.2. Every intuition that affirms truth value is a feeling of correctness. $(∀x)(Ix\to Cx)$ 7.2.3. The feeling ...
Izumi Shinichi's user avatar
4 votes
1 answer
176 views

Mixing predicate logic with propositional logic

In this paper in legal philosophy (https://doi.org/10.3790/rth.40.1.1) on p. 25, the author writes: P(x) ↔ (x → ¬h) This is supposed to be a formalisation of the following statement: "Every ...
Pascal Meier's user avatar
5 votes
3 answers
248 views

How is "so" correctly translated into predicate logic?

I have come across an exercise that asks to have “There is only one ball, so you need to have it” translated into predicate logic. Using the predicates $\text{Ball}(x)$ for $x$ is a ball and $\text{...
CaptnBanana's user avatar
2 votes
3 answers
106 views

Why is implication used instead of conjunction in this instance of translating English into a logical statement?

Can you please help me understand why implication was used instead of conjunction in the answer to this practice question? I have been struggling with nested quantifiers and when to use implication ...
want2understand's user avatar
1 vote
1 answer
63 views

Translating an English statement into a nested quantification with 3 variables.

I am trying to translate: "There is a student in this class who has been in every room of at least one building on campus." My solution was: $$ \exists x \exists b \forall r(B(x, b) \implies ...
want2understand's user avatar
0 votes
1 answer
125 views

Problems with nested Predicate in FOL

In my teacher's lecture, its have a problems like this: execute statements based on the following base predicates $L(x)$: $x$ is a logician $f(x)$: a function that return values is a friend of $x$ ...
chews's user avatar
  • 23
1 vote
1 answer
40 views

Translating English to predicate logic with units

I am having difficulty translating this fragment from a larger sentence into predicate logic: three pets bathe together. Let pets be P(x) and B(x,y) be bathe togehter. How would I deal with the number ...
CSwizard's user avatar
0 votes
1 answer
128 views

"More than two" in First Order Logic

I'm trying to express the following statement which has "More than two" in first order logic. More than two people work at Google Using the following: Name: Google Property: People Relation: ...
TheCoder's user avatar
0 votes
1 answer
54 views

Quantifiers in first order logic

I am trying to express the following sentence in first order logic with quantifiers: All cats like to eat any mouse Using the following properties and relation: Properties: Cat, Mouse Relation: ...
TheCoder's user avatar

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