# Tagged Questions

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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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### 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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### Finding Truth Values Of Nested Quantifiers

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 ...
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### Determining The Truth Value Of Quantified Statements

The problem I am working on is: Determine the truth value of each of these statements if the domain consists of all integers. a) $∀n(n+1>n)$ b) $∃n(2n=3n)$ c) $∃n(n=−n)$ ...
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### How to use the Rules of Inference to a statement from two premises

The problem is as follows: Given the premise ∀x(P (x) ∨ Q(x)) and ∀x((¬P (x) ∧ Q(x)) → R(x)) is true, use the rules of inference to show that ∀x(¬R(x) → P(x)) is also true. (The domains of all ...
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### Quantified Statements To English

The problem I am working on is: Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people. a) $∀x(C(x)→F(x))$ ...
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### Quantifiers, predicates, logical equivalence

I am asked if $(\exists x) (P(x) \rightarrow Q(x))$ and $\forall x P(x) \rightarrow \exists xQ(x)$ are logically equivalent. I don't think they are but how will I prove it? Am I supposed to use ...
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### Converting $\exists x \exists y (x\geq y)$ into English

$\exists x \exists y (x\geq y)$ The universe of discourse is all real numbers. This says that there exists an $x$ and there exists a $y$ such that $x\geq y$. But what is this actually trying to say? ...
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### Are these statements logically equivalent? (quantifiers)

Is $\forall x(P(x) \vee Q(y))$ the same as $(\forall x P(x)) \vee Q(y)$? I understand that if I had $\forall x(P(x) \vee Q(x))$, that it is not the same as $(\forall x P(x)) \vee (\forall x Q(x))$. ...
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### nested quantifiers

In the domain of integers, $P(x,y)$. predicate "$xy = 12$" I'm not sure why $(\forall x)(\exists y)P(x,y)$ is false statement. "For all $x$, there are some $y$, such that $xy = 12$". ex.: ...
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### Why is it true that $∀x((-x)^2=x^2)$?

I'm trying to learn discrete math and I'm lost as to why this truth value is true. Can anyone please explain why? The domain consists of all real numbers. $∀x((-x)^2=x^2)$ The answer is True, but I ...
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### Express each of these sentences in terms of $Q(x, y)$, quantifiers, and logical connectives,

Let Q(x,y) be the statement “x has been a contestant on quiz show y”, where the domain of x is the set of students and the domain for y consists of all quiz shows. For each of the English sentences ...
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### Are these two statement equivalent?

$\forall x \exists y P(x,y)$ $\exists x \forall y P(x,y)$ where P(x,y) means x is smaller than y. I believe that they mean the same thing.
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### Are these statements equivalent (quantifiers)?

$\neg \forall x \exists y \neg P(x,y)$ is equal to $\exists x \exists y \neg P(x,y)$ I had to make sure, because I wasn't sure at all.
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### Is this equivalence true?

Is this equivalence true? $(\forall x (P(x)) \wedge (\exists y Q(y)) \equiv \forall x \exists y(P(x) \wedge \exists x Q(y))$ Here is what I did so far. If the LHS is true, then there exists a x ...
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### What's the negative statement of

$\forall x: P(x) \rightarrow Q(x) \vee R(x)$ Is it for some $x$, if not $P(x)$, then not $Q(x)$ and not $R(x)$?
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### Is double quantifying a variable possible in predicate logic?

I read it as "Everyone is either a student or has read every book". But what's the use of the existential y outside the bracket? ...
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### Are these two predicate statements equivalent or not?

$\exists x \forall y P(x,y) \equiv \forall y \exists x P(x,y)$ I was told they were not, but I don't see how it can be true.