Let K be a subset of $\mathbb{R}$ (The real numbers)
Statement:
John likes $K$ if and only if $∃ a∈\mathbb{R}$ such that $∀ x∈K , a ≤ x$
Question: Does John like all subsets of the real numbers?
My thinking: I think that john does like all subsets of the real numbers, even if the subset $K$ was to be an empty set. This is because the if the subset was to be an empty set, there would be no elements in the empty set, thus making it vacuously true. so that would mean john does like all subsets of the real numbers.
Is this correct, Does john like all subsets of the real numbers, even if the subset was an empty set?
John indeed likes the empty set. Your reasoning about that is correct. But he does not like all subsets of $\mathbb R$. He only likes subsets that have a lower bound in $\mathbb R$. E.g. he does not like $\mathbb R$.