Questions tagged [quantifiers]
The quantifiers $\forall$ ("for all") and $\exists$ ("there exists") distinguish predicate calculus from propositional logic.
1,904 questions
1
vote
1
answer
76
views
Quantifier distributivity dilemma?
From the distributive property of quantifiers, we know,
$$\forall\ x\in\ X,\left(P\left(x\right)\land Q\left(x\right)\right)\equiv\left(\forall x\in X,P\left(x\right)\right)\land\left(\forall x\in X,Q\...
- 33
0
votes
1
answer
106
views
Justifying universal generalization after existential elimination using dependence between quantifiers
In an attempt to transform following equivalent definitions of the cartesian product from the first into the second, trying to be rather formal about it:
$$
\forall A : A \in X \times Y \...
- 21
1
vote
1
answer
128
views
Translating "Someone has visited every country in the world except Libya"
Let $V(p, c)$ mean that person $p$ has visited country $c$ in the world.
Is the following deconstruction correct?
Someone has visited every country in the world except Libya.
There is a person $p$ ...
- 27.8k
1
vote
2
answers
130
views
Translating "Some triangles are green" [duplicate]
I have a pretty basic but tricky question about predicate logic.
$T(x) : x$ is a triangle
$G(x) : x$ is green
Using the above predicates, I am pretty sure that the translation of the sentence "...
1
vote
2
answers
347
views
Vacuous truth in the definition of differentiability
I have some knowledge of what vacuously true means: almost every case can be simplified as
$\forall x$ $P(x) \Rightarrow Q(x)$, if $P(x) = \bot$, then $P(x) \Rightarrow Q(x)$ is always true no matter ...
- 586
10
votes
1
answer
454
views
Is this correctly worded? "If there exist integers $m$ and $n$ such that $12m + 15n = 1$, then $m$ and $n$ are both positive."
In "A Transition to Advanced Mathematics", 8th edition, by Smith, Eggen and St. Andre, exercise 1(f) in chapter 1.6 on page 60 says:
Prove that
if there exist integers $m$ and $n$ such ...
- 758
1
vote
1
answer
84
views
Topological Equivalence
I'm currently looking at when two topologies $\tau(\mathscr B)$ and $\tau(\mathscr B')$ generated by bases $\mathscr B$ and $\mathscr B'$ are identical, and for this, we have:
$$\tau(\mathscr B) = \...
- 1,097
2
votes
3
answers
294
views
Understanding the scope of quantifiers
I’m working through Example 2.3 in a logic text 1, which presents the formula
$$
\forall x.\;p\bigl(f(x), x\bigr)\;\to\;\Bigl(\exists y.\;p\bigl(f\bigl(g(x,y)\bigr), g(x,y)\bigr)\;\land\;q\bigl(x, f(...
- 395
-1
votes
1
answer
65
views
A problem about formalization of an argument with First Order Predicate Logic [duplicate]
I have task regarding formalization in first order predicate logic.
The natural language goes:
"Peter Petson is either completely tone-deaf or is listening to Professor Charles. Everyone who ...
- 35
3
votes
1
answer
172
views
$ ∃x \:(Φ(x) ∨ Ψ(x))$ versus $∃x \:∃y\: (Φ(x) ∨ Ψ(y))$
I'm learning about moving quantifiers to the front of logic formulas (prenex normal form). I have this formula:
$$∃x\, Φ(x) ∨ ∃x\, Ψ(x)$$
My conversion attempts have produced two different results, ...
- 61
2
votes
2
answers
451
views
Do we need to follow the quantifier order when applying a theorem/lemma?
TL;DR: I understand that the order of the quantifiers is important in understanding mathematical statements, but it seems that we can totally ignore this order when applying mathematical statements? I ...
0
votes
2
answers
241
views
The logical equivalence of two statements [closed]
Suppose that $(R,\prec)$ is a totally ordered set, and let $S\subseteq R.$ Define:
$$P:=\forall x\in S \ \forall y\in R(y\preceq x\Rightarrow y\in S),$$$$Q:=\forall x\in S (\forall y\in R(y\preceq x)\...
- 17.8k
0
votes
0
answers
57
views
Implication versus conjunction [duplicate]
So I was going through the Rosen's book on Discrete Mathematics and I stumbled upon an example which had me confused from hours.
This was the question:
Consider these statements, of which the first ...
- 11
0
votes
1
answer
142
views
Intersection of set with arbitrary intersection PROOF
How do we prove:
$$\left(\bigcap_{i\in I} L_i\right)\cap R=\bigcap_{i\in I}\left(L_i\cap R\right)$$
What we want to prove is that:
$$ ~(x\in R)\text{ and }(\forall\ i \in I ~~ (x\in L_i))~\iff\forall\ ...
- 399
6
votes
1
answer
372
views
Are there logics where you can quantify over quantifiers?
I'm beginning to work through William Weiss and Cherie D'Mello's Fundamentals of Model Theory. I noticed that one of the proofs in Chapter 0 handles the following two cases where the only difference ...
- 329