Find a model M where $M \models(∀x)(∃y)R(x,y) ∧ ¬(∃y)(∀x)R(x,y)$

Question

Find a model M where $M \models (∀x)(∃y)R(x,y) ∧ ¬(∃y)(∀x)R(x,y)$

I'm not sure about what does this sentence mean. I was thinking the first half part as for all $x$, there exists $y$ such that $x R y$.

Answer 1

Hint: Consider $\mathbb{N}$ with $R$ interpreted as $<$.

Answer 2

Your interpretation of the first part is correct.

The second part says that there is no $y$ such that $xRy$ for every $x$.

Answer 3

Find a model $\mathscr M$ where $\mathscr M \models (∀x)(∃y)R(x,y) ∧ ¬(∃y)(∀x)R(x,y)$

The antecequent reads: "All things are R-related-to something but there is nothing R-related-by all things."

Answer 4

A model is a domain together with an interpretation of the predicates (and constants, but we have none here). So consistent with Kyle's answer, think of the fact that every natural number, 0, 1, 2, ... has something greater than it, but no natural number is greater than every number.