# Tag Info

Accepted

### In proofs, are "for each" and "for any" synonyms?

Compare A1. If there’s a simple solution for each of the problems, the test is too easy. A2. If there’s a simple solution for any of the problems, the test is too easy. These are not equivalent. ...
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### "If everyone in front of you is bald, then you're bald." Does this logically mean that the first person is bald?

You can see what's going on by reformulating the assumption in its equivalent contrapositive form: If I'm not bald, then there is someone in front of me who is not bald. Now the first person in ...
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### "If everyone in front of you is bald, then you're bald." Does this logically mean that the first person is bald?

Mathematical logic defines a statement about all elements of an empty set to be true. This is called vacuous truth. It may be somewhat confusing since it doesn't agree with common everyday usage, ...
• 42.4k
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### Does the unique existential quantifier commute with the existential quantifier?

No, they do not commute. Consider for example the nonnegative reals as a linear order. Then $$\exists x\exists !y(y\le x)$$ is true (take $x=0$), but $$\exists!y \exists x(y\le x)$$ is false since ...
• 244k

### In proofs, are "for each" and "for any" synonyms?

They are synonymous, but may be used in different contexts.   Both declare that the predicate applies to every entity in the domain.   However, "for each" is more often used in an imperative ...
• 129k

### Does leaving universal quantifiers up to context lead to ambiguity?

Yes, it does generate ambiguity. Let $E$ and $O$ be predicates for "is even" and "is odd," respectively, and consider the difference between $$\forall x(E(x)\rightarrow O(x))$$ and ...
• 244k

### Is the Lyapunov stability definition ambiguous?

You're not interpreting the logic correctly. It doesn't say “there exists a single positive number $\delta$ which works for all positive numbers $\epsilon$”, it says “for any positive number $\epsilon$...
• 53.1k
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Accepted

### Why does universal generalization work? (the rule of inference)

1. What is meant by the second paragraph ? That is, when we assert from ∀xP(x) the existence of an element c in the domain, we have no control over c and cannot make any other assumptions about c ...
• 45.7k
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### Example of a quantifier which is not compatible with ordered pairs.

How about $\exists!$, the unique existence quantifier? Working in the structure $(\mathbb{N};<)$, we have that $\exists !x\exists !y(x>y)$ is true but $\exists !(x,y)(x>y)$ is obviously false....
• 244k
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### Scope of quantifier (LOGIC DISCRETE MATH)

It's often a matter of parentheses: In a formula like $\forall y P(y)$ the $y$ in $P(y)$ is within the scope of the $\forall y$, but in a formula like $\forall y Q(x) \land P(y)$ it is not, since ...
• 99.6k
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### How to understand intuitively why "$\exists !x \exists !y$" is not equivalent to "$\exists !y \exists !x$"?

To start things off, let's whip up a simple-to-understand example showing that the sort of switching you have in mind can't possibly be legal in general (it sounds like you already know this, but ...
• 244k

### What's the difference between $∀x\,∃y\,L(x, y)$ and $∃y\,∀x\,L(x, y)$?

That's a great example of why quantifiers don't commute! For the sake of simplicity, assume everybody in the world is married, and everybody loves his spouse. Then the first formula is satisfied. ...
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### Why is quantifier elimination desirable for a given theory?

You're quite right that we can "shoehorn in" quantifier elimination to any theory we want, by adding new predicates for all old formulas (this is called Morleyization if I recall correctly). So ...
• 244k

### What is the inverse for ∀

Neither of the sentences you've written does the job. First of all, "$\forall x(F(x)\wedge P(x))$" is absurdly strong: forgetting the $P$-part it implies that everyone is your friend, which ...
• 244k
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### Is the Lyapunov stability definition ambiguous?

$ϵ$ is primary, $δ$ is secondary depending on it. As you have chosen $x(0)$, you already know the radius $δ$ in $\|x(0)-x_e\|<δ$ and thus also the $ϵ$ it is based upon. So there remains no freedom ...
• 126k
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### Difference between “for some $k$” and “for some arbitrary $k$”

In your first example, I would just write “where $k$ is an integer”. But “some” is okay. The point is, since $n$ is already known, $k$ is completely determined. Writing “where $k$ is some arbitrary ...
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