# Tagged Questions

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### Deciding between implication and conjunction

This is one of the solved problems in Velleman's How to prove book: Analyze the logical forms of the following statements: 1) John likes exactly one person. ...
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### Analyzing the logical form of “All married couples fight”

This is one of the example problems in Velleman's How to Prove book: Analyze the logical forms of the following statements. All married couples have ...
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### Concluding Truth Value from Universe of Discourse

I have been working on the following problem from Velleman's How to Prove book: Are these statements true or false? The universe of discourse is the set of ...
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### Idiomatic mathematical english statement for ∃x[P(x) ∧ ∀y(P(y) → y ≤ x)]

I have been working on problems from Velleman's How to Prove book and hit upon the following problem: Translate the following statements into idiomatic ...
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### Questions regarding Universal Quantifiers

The question is to show that: $$\exists x:(P(x) \implies Q(x))\qquad\equiv\qquad\forall x:P(x) \implies \exists x:Q(x)$$ First I use double negation to get to the universal quantifier since it ...
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### Understanding logical form of “Nobody in the calculus class is smarter than everybody in the discrete math class”

I'm self studying How to Prove book and have been working out the following problem in which I have to analyze it to logical form: Nobody in the calculus class ...
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### Analyzing Logical Forms involving quantifiers

I have been solving the following problem from How to Prove book: Analyze the logical forms of the following statement: Everyone likes Mary, except Mary ...
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### Relation between existential and universal quantificator in category theory

Let $\mathscr C$ be a cartesian (i.e. with finite limits) category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$ (here $I$ denote the terminal object). Let $f:X\to Y$ and ...
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### 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 ...
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### Universal and Existential Quantifiers

Are there any examples of a predicate P(x ) of a variable x such that the truth value of P(x) remains invariant under exchange of the Universal Quantifier ∀ and the Existential Quantifier ∃ -thanks
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### Definition and decidability of bounded quantifiers

Consider quantifier-free formulas $P(x,y) = Q(x,y)$ of Peano arithmetic. Consider $P(x,y),Q(x,y)$ to be terms composed of variables $x,y, \operatorname{succ}, +, \times$. Note that these are ...
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### Can universal instantiation be used more than once?

I'm trying to follow a proof in a logic text and it seems like the author used universal instantiation twice to reach the needed result. I was under the impression that you could only use UI one time ...
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### proof of quantifier elimination in theory of real closed field/reals and existential quantifier over atomic formula

Standard proof of quantifier elimination for theory of real closed field/reals uses induction, as in Wikipedia article (http://en.wikipedia.org/wiki/Quantifier_elimination#Basic_ideas). However, it ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Semantics and Logical structure in Definitons

Continuation of Free and bound variables in "if" statements definitions: A number is even if it is divisible by $2$. The number is even if it is divisible by $2$. Is the usage of the ...
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### Free and bound variables in “if” statements

The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion $\forall x(x>2)$ is false. In the first case ...
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### Logical structure of definitions

Here are some "concepts" that are confusing me: The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion ...
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### $\exists \implies \forall$

I want to see some example theorem, when existence implies universality, so $\exists \implies \forall$ is true. I think matematical induction is a related technique, but I just don't see that ...
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### Prove $\neg\exists x \neg P(x)$ is a logical consequence of $\forall x P(x)$

How would you prove that 2. is a logical consequence of 1. using a Fitch style proof? $\forall x P(x)$ $\neg\exists x \neg P(x)$
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### Proof of Drinker paradox [duplicate]

I searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...
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### Truth set of a Universal Quantifier and Family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A(A\in F \implies x\in A)\bigr\}$$ If my understanding serves me correctly this notation means that all the $x$ ...
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### Definition of intersection of a family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A: (A\in F \implies x\in A)\bigr\}$$ If my understanding serves me correctly this notation means that all the $x$ ...
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### Difference between bound and free variable

What is the difference between $\forall x (P(x)\implies Q(x))$ and $P(x)\implies Q(x)$ I know in the first one the variable x is bound but in the second one the variable is free. What are the ...
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### Symbolic logic proof

Can any one please give me the correct proof for this, i got this far but i am stuck. Thank you!
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### Domain of discourse and quantifying in predicate logic

I am struggling with an idea about how quantifiers relate to domains of discourse. Given a statement "$x$ is divisible by $2$" represented by the predicate $D(x)$,the predicate currently has no truth ...
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### Is there way to classify the quantifier rank $m$ first order sentence in first order logic

In its simplest situation, for example, if the signature contains only a binary relation $\sigma$, so the signature $\tau = \{ \sigma \}$, what are the inequivalent classes of all first order ...
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### Commutivity of unique existence quantifiers

Find an expression P(x,y) to disprove the following equivalence, $(\exists!x)(\exists!y)P(x,y)\Leftrightarrow(\exists!y)(\exists!x)P(x,y)$ I could only think of a few statements of $P(x,y)$ that ...
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 ...