Given the following dictionary, how would the sentences below be translated in to a language using quantifiers? My attempts are shown as well:
Dictionary:
$L$: a two place predicate which means -larger than-
$N$: a one place predicate meaning -is a number-
$a$ = -zero-
How would I symbolize the following sentences?
No Number is Larger than all numbers:
Attempt: $\neg \exists x\forall y(N(x)\wedge N(y)\wedge L(x,y)) $
OR would it be: $\neg \exists x\forall y\big(N(x)\to (N(y)\wedge L(x,y))\big)$.
Some Numbers Larger than Zero are larger than some numbers larger than zero:
Attempt: $\exists x\forall y\Big(\big((N(x)\wedge L(x,a))\wedge(N(y)\wedge L(y,a))\big)\to L(x,y) \Big)$
What is the correct answer to the two above?
"No Number is Larger than all numbers:"
When saying "for any element with property P thing Q should happen" we need to use $\forall x$ and implication. If we do not, and instead use $\wedge$ together with $\forall x$ we will say that all elements satisfy both property $P$ and property $Q$ So your first attempt $\neg \exists x \forall y (N(x)\wedge N(y)\wedge L(x,y))$, the inner part of the formula say that $x$ and $y$ are both numbers and $x$ is larger than $y$. As $y$ is bound by a $\forall$ quantifier this means that all elements are numbers, which may not be what we intended to say.
Your second attempt is better using an implication. However, the implication is in the wrong place. As it stands it say $N(x)$, where $x$ is bound by the existential quantifier. Thus again if we can find an element which is not a number then $N(x)$ will be false, thus the implication will be true and the whole formula will be false. Instead the formula should look like
$$\neg \exists x\big(N(x)\wedge \forall y (N(y)\rightarrow L(x,y))\big)$$ Here we clearly state that $x$ is a number, and that if $y$ is a number then $x$ is larger than $y$. Note that there are many equivalent formulations of this formula, thus if this is an exercise from a textbook, then the book-answer may look different.
"Some Numbers Larger than Zero are larger than some numbers larger than zero:"
This sencent only state "some" multiple times, thus there is no need here to involve $\forall$ quantifiers instead we just spell it all out like this:
$$\exists x\exists y\big(N(x)\wedge N(y) \wedge L(x,y) \wedge L(x,a)\wedge L(y,a)\big)$$