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

learn more… | top users | synonyms

1
vote
1answer
28 views

Reasoning informally about $\{x \in B \mid x \notin C\} \in \mathscr P(A)$

Attempting to apply more flexible, informal reasoning to predicate logic as demonstrated helpfully to me by another user in answer to my last question. $\{x \in B \mid x \notin C\} \in \mathscr P(A)$ ...
3
votes
2answers
22 views

Rewriting $\mathscr P(\bigcup_{i \in I} A_i)\not\subset\bigcup_{i \in I} \mathscr P(A_i)$ in more fundamental terms.

Working through Velleman's "How to Prove It" and currently having a bit of difficulty with a problem where I'm asked to rewrite this: $$\mathscr P\left(\bigcup_{i\in I} A_i\right)\not\subset\bigcup_{...
0
votes
1answer
28 views

How can I express each of these quantifications in English?

Let T(x) be the statement "x has visited Tashkent" where the domain consists of all students of my school. How can I express each of these quantifications in English? ...
1
vote
2answers
33 views

What is the negation of ∀x∃y¬P(x,y) without using ¬?

Found it to be ∃x∀yP(x,y). Is this right?
1
vote
0answers
31 views

Negation of double universal quantifications

In logic, when I want to negate the formula $$\forall x \forall y( F(y) \land A(y) \to \neg G(x,y))$$ what is the correct equivalent? Intuitively, I think it gives $$\exists x \forall y (F(y) \land ...
1
vote
2answers
26 views

Is this conclusion via rules of inference correct?

Use rules of inference to show: ∀x(P(x) → Q(x)) premise ∀x(Q(x) → R(x)) premise ¬R(a) premise ¬P(a) conclusion I have a lot of trouble with these sort of questions and was wondering if I did this ...
0
votes
2answers
21 views

Presenting well a sentence with quantifiers

What is the syntax rule to present a syntax with quantifiers ? Should we rather write : \begin{equation} \forall x\in \mathbb N\quad \exists y\in\mathbb N \quad x<y\qquad (1) \end{equation} or \...
0
votes
1answer
16 views

Nested Quantifiers (And vs Implies)

I would like to understand something regarding the nested quantifiers in discrete math. In the following question part (c): Let $M(x,y)$ be "$x$ has sent $y$ an e-mail message", and $T(x,y)$ be "$...
0
votes
4answers
87 views

why is $\forall x (p(x) \implies q(x)) \not\equiv (\forall x p(x)) \implies (\forall x q(x))$

I'm having a hard time wrapping my head around why $\forall x (p(x) \implies q(x)) \not\equiv (\forall x p(x)) \implies (\forall x q(x))$
2
votes
3answers
86 views

What does a period in between quantifiers mean?

I'm currently reading the notes (rather a book) of an MIT preliminary math course for discrete mathematics. In section on page 39, some ZFC axioms are written and roughly explained. For example, the "...
1
vote
1answer
46 views

Compound Quantifier

Can any one help me what will be the universe discourse of these two statements? if both statement has natural numbers or same universe of discourse what will be values, that makes 1st statement true ...
0
votes
1answer
12 views

Decidability of quantifier-free formulae in Peano- and True Arithmetic

It is well-known that validity in Peano Arithmetic is undecidable. It is less well-known that validity is already undecidable in True Arithmetic (the theory of the standard model of Peano Arithmetic). ...
1
vote
1answer
18 views

Existential and Universal Quantifiers

Quantifiers (a) Please see below. I cannot work out why one is correct. If $x < 0$, then there's no value $y \in \mathbb{R}$ so that $y^2 = x$. (b) If I have $\exists$ followed by $\forall$, then ...
0
votes
1answer
18 views

Definition of rational number in logical expression format

So I have to translate the following definition of rational number into logical expression. The real number r is rational if there exist integers p and q with q = 0 such that r = p/q. I have ...
2
votes
1answer
23 views

How to express the following statement with Quantifiers and Predicates

Use quantifiers and predicates with more than one variable to express this statement: There is a student in this class who has taken every course offered by one of the departments in this school ...
0
votes
0answers
61 views

Is there any linear algebra textbook presented using logical symbols?

I'm currently going through a book called Linear Algebra Done Right by Axler, and to be honest, his book seems to be very loose with what things he defines. For instance , the symbol 0 could be mean a ...
1
vote
3answers
48 views

logic: order of quantifier with free variables

Take the sentence, "You can't win them all." This could be logically written as "For all people, there exists a thing they cannot win at." $\forall x.\exists y.(\neg win(x,y))$ Now suppose I was ...
0
votes
0answers
26 views

Negation of a quantified statement regarding an implication.

I am trying to improve my understanding of the negation of a quantified statement where the statement is an implication. I am doing a practice problem which I dont have the answer to from the textbook ...
1
vote
1answer
19 views

Question on negation of a quantified statement

There is a slight confusion I am having when comparing my answer to a solution for a problem. Basically, the question asks me to state the negation of For every integer $n$ such that if $n$ is ...
2
votes
2answers
44 views

Is the universal quantification symbol $\forall\;$ known as “for any x” or “for all x” in First Order Logic & why is this different to Discrete Math?

I'm reading a book by Mark Tarver called Logic, Proof and Computation. Chapter 8 (starting p71) is about First Order Logic. On page 77 (of Chapter 8) the author writes: For any value for 'x', in ...
1
vote
1answer
57 views

Quantifier elimination, $(\mathbb{R}, <)$

I have a general question about quantifier elimination. Which kinds of formulas do you have to observe? For example let T be the theory of $(\mathbb{R}, <)$ and I want to show, that this theory ...
1
vote
1answer
28 views

Converting ∃ to ∀ and vice versa

I'm having some trouble getting my head around the conversion of quantifiers. For instance, I know that $\forall x \,F \,\equiv\, \neg\exists x \, \neg F$ and conversely. $\exists x \,F \,\equiv\, \...
0
votes
0answers
20 views

Symmetry and transitivity with the existential quantifier

I can't find any resource that would indicate whether symmetry and transitivity relations are possible or not with the existential quantifier or when quantifiers are combined. I'm interested in the ...
0
votes
1answer
30 views

Quantifier in Set definitions

Can the definition be made more readable: $\overline{R_1} = \{ (j_1,j_2)\mid j_2 < j_1 + \Gamma^r_a + c^r_w \text{ and } j_1 \in J^r_v \text{ and } j_2 \in J^r_v \text{ and } a = (v,w) \in A^r ...
0
votes
3answers
53 views

Question about universal quantifier

when I was reading a paper about the universal quantifier, I met this equation, says we can do conversions like the following: A -> B ≡ ¬A v B can anyone help ...
2
votes
1answer
21 views

true or false statements based on predicate logic

Q) Let $P(x,y)$ be the predicate $y=2x$. Consider the statements a)$\forall x \exists y P(x,y)$ b)$\forall y \exists x P(x,y)$ c)$\exists y \forall x P(x,y)$ where $x$ and $y$ range over the ...
1
vote
2answers
38 views

Is this statement logically true? If so How?

Q) Is the statement (∃xQ(x)∧∃xR(x))↔∃x(Q(x)∧R(x)) logically true? If it is, explain why. If it is not give an interpretation under which it is false. I have asked this previously but did not get an ...
1
vote
1answer
33 views

translating uniqueness quantifier algorithmically

Given a claim with a uniqueness quantifier $\exists$!, such as: $$\forall x \exists!y \ P(x,y) $$ A standard translation that uses $\forall$ and $\exists$ only (there are several possible ones) is: ...
-2
votes
1answer
25 views

Proving/disproving statements with a given context of natural numbers.

How do I prove the following statements or their negations in the context where $x$ and $y$ are rational numbers in the closed interval $[-\sqrt{2}, \sqrt{2}]$? Statement 1: $\forall x \exists y\; x &...
1
vote
2answers
25 views

Predicate logic: Symbolize a sentence using a dictionary and two-place predicates

Given the following dictionary, how would the sentences below be translated in to a language using quantifiers? My attempts are shown as well: Dictionary: $L$: a two place predicate which means -...
1
vote
1answer
73 views

Alternatives to pure quantifier logic

Are there some alternatives for pure quantifier logic? Pure quantifier logic is axioms and rules of inference added to proposition logic to result first order logic. Are there other axioms that ...
0
votes
2answers
26 views

Trouble understanding negation of definition of convergent sequence.

Definition of convergent sequence: $$\forall \varepsilon >0, \exists N \in \mathbb R: \forall n \in \mathbb N \ (n \ge N \implies d(x_n,x) < \varepsilon)$$ I found the negation to be: $$\exists ...
0
votes
0answers
19 views

Composing relation with identity yields original relation

Let $\phi: F\rightarrow G$ be a relation and $id_F$ be the identity relation on $F$. Then $\phi\circ id_F = \phi$ . Attempted proof: $$\phi\ \circ id_F = \{(f,g)\in F\times G: \exists f_2\in F((f_1,...
1
vote
2answers
27 views

Prove $\forall X [X \subseteq A \land y \in A] \rightarrow \exists X[X \subseteq A \land y \in X]$

Prove: $\forall X [X \subseteq A \land y \in A] \rightarrow \exists X[X \subseteq A \land y \in X]$ proof: Proving by contradiction, suppose $$\forall X [X \subseteq A \land y \in A] \text{ and } \...
0
votes
1answer
39 views

quantifiers that bind nothing

Consider $\forall x \forall y P(y,y)$, where $x$ is quantified, but does not appear. Is quantifying a variable that otherwise does not appear well-formed or meaningful? I've seen this sort of thing ...
1
vote
1answer
29 views

Discrete Mathematics - How to Express the phrase “There is no one who did action y”

Let's say $P(x,y)$ means $x$ sent an e-mail to $y$. If I want to say that no one has sent a message to Jean, then aren't there multiple ways to do this? $\neg Ǝx(P(x, Jean))$ But I can also say $∀...
0
votes
1answer
36 views

conditional proof with different variable [closed]

I don't know how to prove $(Ǝx~F(x)~→~Ǝx~G(x))$ with conditional proof from: $(Ǝx~F(x) ~→~ ∀z~H(z))$ $H(a)~→~G(b)$ I have a problem with different variables here and conditional proof as well.
1
vote
0answers
11 views

How to interpret this index variable and corresponding summation limits?

Let $TD$ be an integer (That is, $TD \in \{ 1,2,3,4,\dots \}$). I am looking at the following statement: If $TD$ is even, the (solution) is given by $$ \forall \frac{TD}{2}\in \mathbb{N}: x_i^* = 1 - ...
1
vote
3answers
39 views

Simple proof involving quantifiers

Task: Prove this theorem: $ \exists x (P(x) \Rightarrow \forall yP(y) )$. I got this far: I figured out this is equivalent to $\exists x (\neg P(x) \lor \forall yP(y) )$. I don't understand ...
0
votes
2answers
38 views

How do I translate 'no philosopher student admires any rotten lecturer' into quantificational logic formula?

Let's assume that $Fx=x$ is a philosophy student, $Rx=x$ is a rotten lecturer, and $Mxy=x$ admires $y$. My translation of the sentence was $\forall x(Fx\supset\neg\forall y(Ry\supset Mxy))$, but my ...
3
votes
3answers
166 views

B is an element of some power set of A such that A is an element of F.

$$B \in \{\,\mathcal P(A) \mid A \in F\,\}$$ I can't quite figure this out. My textbook says this statement is equivalent to $$\exists A \in F\ \forall x\ (x \in B \leftrightarrow \forall y\ (y \in ...
5
votes
1answer
55 views

correct typesetting for quantifiers

For years I have been typing and writing quantifiers in a certain way. Now that I am writing my thesis, my adviser is taking issue with some of these things. Since he is my adviser I'm going to do ...
1
vote
1answer
26 views

How can I interpret a multiply-quantified statement?

∃ x ∈ R such that ∀ y ∈ R, x + y = 0. Can anyone help me rewrite this statement in plain english without symbols or variables? So far I have "There exists a real number whose number and other number ...
41
votes
11answers
9k views

“If everyone in front of you is bald, then you're bald.” Does this logically mean that the first person is bald?

Suppose we have a line of people that starts with person #1 and goes for a (finite or infinite) number of people behind him/her, and this property holds for every person in the line: If everyone ...
1
vote
1answer
20 views

Translating this nested quantifier to english (negation of nested quantifiers)

So i'm a total newb at this so I need help on one the questions for my assignment: Let S(x) = “x is a student at Bronx Community College”; F(x) = “x is a faculty member at Bronx Community College”, ...
1
vote
0answers
32 views

Universal Quantification of a Predicate P(x)

Consider the following statement: Every number is less than its square. Write the statement “Every number is less that its square” symbolically by defining a predicate and using a quantifier. Answer: ...
1
vote
5answers
54 views

How should I prove $\forall x \in \mathbb{Q}$ where $x > \sqrt 2$ , $\exists y \in \mathbb{Q}$ where $\sqrt{2} < y < x$

How should I prove the below statement? $\forall x \in \mathbb{Q}$ where $x > \sqrt 2$ , $\exists y \in > \mathbb{Q}$ where $\sqrt{2} < y < x$ I tried to prove it by contradiction ...
1
vote
4answers
37 views

Universal Quantifiers [duplicate]

What is the difference between ∀x∈U : ~p(x) and ~∀x∈U : p(x) ?? Could anybody give any English sentences explaining both of them?...
0
votes
0answers
33 views

Did I correctly translate 'Whoever loves Myfanwy, loves a philosopher only if the latter loves Myfanwy too.' into quantificational logic?

There is an exercise in my textbook. Suppose ‘$m$’ denotes Myfanwy, ‘$n$’ denotes Ninian, ‘$o$’ denotes Olwen, ‘$Fx$’ means x is a philosopher, ‘$Gx$’ means x speaks Welsh, ‘$Lxy$’ means $x$ loves ...
0
votes
1answer
44 views

sets with quantifiers

My professor wrote the following theorems and definitions on the blackboard. ...