Use logical quantifiers to write: "Everybody loves somebody sometimes" (Where U=all people) I came up with this but not sure how to type symbols in here.
$$\forall x \in U\,: \exists y\in U: x \text{ loves } y.$$
So... upside down A="For all" Backwards E for "there exists" curly little e for "belongs to" My apologies as I don't know how to insert symbols like those. Am I on the right track to this? Or is this not even close? Any help appreciated.
On
$\forall Body(1)\; \exists Body(2)\; \exists Time(t)\; (Body(1)\ne Body(2)\land Body(1)\; loves\; Body(2)\; at \;Time(t)).$
On
With apologies to Dino, that sentence is a bit ambiguous. Here is one way to look at it:
$\forall x:[P(x) \implies \exists y: \exists t:[P(y) \land T(t) \land L(x,y,t)]]$
where
$P(x)$ means $x$ is a person
$T(t)$ means $t$ is an instant in time
$L(x,y,t)$ means $x$ loves $y$ at instant in time $t$
You're almost correct except the first : or 'such that' isn't needed. $$ \forall\; x\in U\;\exists \;y\in U : x \text{ loves } y $$ another way to write this is $x\in U\implies \;\exists y\in U: x$ loves $y$.