3 votes

The converse of Euclid lemma : counter example?

Euclid's lemma states that if $p$ is prime, then $p \mid ab$ implies $p \mid a$ or $p \mid b$. The lemma makes absolutely no comment on the case when $p$ is composite. In fact, the statement can be ...
kipf's user avatar
  • 1,542
2 votes

Can negations be arbitrarily added to both sides of an equation?

The "first negation law" says that $\lnot \exists x P(x)$ is equivalent to $\forall x \lnot P(x)$ for every formula $P(x)$. The key word here is: every formula. We can replace $P(x)$ by any ...
Ted's user avatar
  • 32.7k
2 votes
Accepted

Is there a simpler Kripke counter-model for this formula?

There isn’t really a nicer model. Any model with finitely many states will fail. To see this, note that maximal elements are “dense” in the finite state case (every element is less than a maximal ...
Mark Saving's user avatar
  • 30.8k
2 votes
Accepted

The definition of proposition and the principle of the excluded third

A proposition is defined as a statement or assertion that can either be true or false. Here's a more direct definition: A proposition/sentence is a string of symbols that is well-formed according to ...
ryang's user avatar
  • 38k
2 votes

The definition of proposition and the principle of the excluded third

A better way to state the first definition is that a proposition is the sort of thing that is eligible to be judged true or false. That is, they are the sort of things $P$ about which it at least ...
Dan Doel's user avatar
  • 3,360
2 votes

There is some integer $n$ such that if $n > 2,$ then $n^2 = 2n.$

Expanding on Robert's observation: This is the given sentence, which is true: there is some integer $n$ such that if $n > 2,$ then $n^2 = 2n$ $\exists n{\in}\mathbb Z \:\big(n > 2\:\to\: n^2 = ...
ryang's user avatar
  • 38k
1 vote

Finding the relationship (equivalence or implication) between two expressions

The second implies the first. Suppose we have the second already. Then we can get in our context the following $x_0 \in X$ and $p(x_0)$ as well as $\forall x \; q(x)$. Combining the first and third ...
AHusain's user avatar
  • 5,009
1 vote
Accepted

Finding the relationship (equivalence or implication) between two expressions

The formula $\exists x Px \wedge \forall x Qx$ states "There exists at least one $x$ such that $x$ is $P$, and for every $x$, $x$ is $Q$." The formula $\exists x [Px \wedge Qx]$ states "...
RyRy the Fly Guy's user avatar
1 vote

Finding the relationship (equivalence or implication) between two expressions

To check the conditional relationship between two sentences, making sense of them is usually faster than formal proof methods: $$\exists X \; (p(X) ∧ q(X))$$ Some object is both pink and quirky. $$\...
ryang's user avatar
  • 38k
1 vote

The converse of Euclid lemma : counter example?

If $p$ is some number that can divide another number $a$, $b$, or both, that fact has no bearing on whether $p$ is prime. All we can say is that $p$ is a divisor of $a$ or $b$. Observe also that the ...
Jam's user avatar
  • 9,475
1 vote

Santa Claus does not exist. Therefore, something does not exist. Valid?

P1: Santa Claus does not exist. C: Something does not exist. P1: Santa Claus does not exist. P2: Santa Claus is something. C: Something does not exist. The above arguments are phrased for maximal ...
ryang's user avatar
  • 38k
1 vote

Why does switching the quantifiers make the statement false?

"For all x in a Set A, x have property P" is same as saying, Given any x in A, x has property p The famous analogue to understand the difference between change of quantifiers is knowing the ...
Praveen Kumaran P's user avatar
1 vote

$\forall x(P(x) \rightarrow Q(x))$ and $(\exists xP(x) \rightarrow \forall yQ(y))$ having different truth values

The first statement says that $P$ implies $Q$ for every $x$. The second statement says that as long as one $x$ satisfies $P$ then all $y$ satisfy $Q$. For example, let $P(x)$ be the statement "x ...
John Douma's user avatar
  • 11.1k

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