Question

We’ve seen that natural numbers are closed under addition. Determine the closure of natural numbers under the other three operations, and give examples to support your answers.

Answer 1

Natural numbers are not closed under subtraction. In fact, any operation $x-y,\quad y>x$ has no solution using natural numbers.

Natural numbers are closed under multiplication. In fact, any operation $x\cdot y$ means to sum $x$ by itself $y$ times, and we already know that natural numbers are closed under sum.

Natural numbers are not closed under division. In fact, if $y$ doesn't divide $x$, any operation $\frac{x}{y}$ has no solution using natural numbers.