Can someone check my answers for me?
e)“Only zero is divided by all natural numbers.”
∀x(∃y(y|x) → (y = 0))
f) ”Each Pythagorean triple involves at least one even number.”∀x∀y∀z((x^2 + y^2 = z^2) → ∀n(x = 2n V y = 2n V z = 2n))
Passive voice is much more difficult to symbolize that active voice.
Suppose the domain is the set of non-negative integers.
‘Some number is divisible by every number’: $\displaystyle \left( {\exists n} \right)\left( {\forall k} \right)\left[ {k|n} \right]$.
‘Every number is divides some number: $\displaystyle \left( {\forall k} \right)\left( {\exists n} \right)\left[ {k|n} \right]$
Now on the domain the first statement is false, no number is divisible by zero.
But on that domain the second statement is true, 1 divides every number.