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Questions tagged [quantifiers]

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

Quantifiers specify the quantity of objects that satisfy a given formula.

The quantifiers $\forall$ (for all) and $\exists$ (there exists) are the most common, but others such as $\exists!$ (there exists a unique) are also in usage.

Only use this tag if your question is about the usage of a quantifier in a formula. Be sure not to use this tag for any question with quantifiers.

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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 does imply that there exists exactly one value…
jhuk
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Symmetric and transitive relation definition problem

Mathematical definitions and notation really confuse me. For example, a definition similar to the following can be found in many textbooks and online: In mathematics, a binary relation R over a set X is transitive if whenever an element a is…
stillenat
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Statements involving quantifiers

I am confused regarding the following; If we have a statement, for example, $$\exists_{x} \in X, \forall_{y} \in Y, x + y = 0.$$ Now, I'm wondering if you could just choose $x$ as $-y$, or do you have to pick a specific value for $x$?
the man
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$∀x(N(x)→∃y∃x(N(y) ∧ x ≥ y))$ - bounded variables

Using this formula I am trying to see which variables are bounded by which quantifiers. $∀x$ universal quantifier bounds all the $x$'s in the formula as they are all within its bracket. However, does that mean I ignore the $∃x$ which is within that…
Iona
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Help with order of quantifiers

I have to say if the following are true or false and why. Can someone check to see if I understand how orders of quantifiers affect the meaning? For every integer x, there exists an integer y such that y>x. True. Whatever integer x is, there will…
ematth7
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Universal Instantiation

I'm a bit confused about how Universal Instantiation works. I read that you shouldn't really plug in any value "a" for x in "For all x P(x)" unless a choice for "a" pops up in the givens but in the following proof they just select any value y. Is…
ChemDude
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Order of Existential/Universal Quantifiers

I just wanted to check my answers for this because I'm still not that comfortable with it. Which of the following statements are true? (i) $(\forall x \in \mathbb R)$ $x+1>x$ (ii) $(\forall x \in \mathbb Z)$ $x^2>x$ (iii) $(\exists x \in…
thisisourconcerndude
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Contrapositive/contradiction of statement with quantifiers

In general how does one formulate a proof by controposition or contradiction for the following general form: $\forall x\exists ! y (P(x)\wedge Q(y) \rightarrow R(x,y))$ Or more specifically: $\forall x\exists ! y (x\geq 0 \wedge y\geq 0 \rightarrow…
Sonny
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Negation of a Statement with Quantifiers -- If Then?

I need to find the negation of a statement on my homework, specifically problem 19 of secton 3.2 in Discrete Mathematics with Applications by Susanna Epp. The problem is as follows: \begin{align} \neg\left(\forall n\in \mathbb{Z},\:n\:\text{is >…
bjd2385
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Is there an implicit quantifier, or is it always an error when one isn't specified?

I have an exercise book from my university which doesn't specify a quantifier. It uses expressions like "here $A$,$B$,$C$ are sets", or "if $x \notin A$ then ..." (it uses $x$ before it is even defined, just out of nowhere). I'm going to need an…
MasterMastic
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express in words without using the symbol N

Express $$\forall\ n\in\mathbb N\ \exists\ m \in\mathbb N: \ n^4 = m^2$$ in words without using the symbol $\mathbb N$.
ashneel
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Quantifier question?

How would I do the following quantifier and their negation No one loves everybody. or could you say : everybody does not love someone? x is all people So in symbolic this would be $\forall x, \exists y,$ x does not love y. and the denial is…
Fernando Martinez
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Interpretation of quantifiers

While studying quantifiers I got all confused with the following explanation about the order of quantifiers. The statement ∀x ∃y, y > x claims that, for any real number x, there is a number y which is greater than it. In the realm of the real…
VCS
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How can I evaluate existential quantifiers in the set theory?

I can't understand why the following conversions are possible. A = {x ∈ N : ∃n ∈ N(x = 2n)} ⋯ ① = {x ∈ N : x is even} ⋯ ② A is the set of all x is in the natural numbers, such that there is at least one natural number n such that x = 2n. In other…
Aspas
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For all x, there exist a y such that 2x-y=0 in the set of non negative integers.

I have the following proposition in the domain of nonnegative integers: $$ \forall x \exists y \cdot(2x-y = 0) $$ I translate it as the following: For all $x \in \{0, 1, 2,3,.... \}$, there is always a $y$ to be found, such that $y=2x$. The answer…
senseiwu
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