Please Login to access more options.
Exercise (Are The Natural Numbers A Subgroup Of The Integers)
Is $\mathbb{N}$ closed under the operation of addition? Is $\mathbb{N}$ a subgroup of $\mathbb{Z}$?
Click to see a solution.
The sum of two positive integers is a positive integer, so $\mathbb{N}$ is closed under the operation of $\mathbb{Z}$ (which is addition). However, the inverse of $2$ under addition is $-2$, but $-2\notin \mathbb{N}$. This shows that $\mathbb{N}$ is not closed under taking inverses, and hence is not a group (so not a subgroup of $\mathbb{Z}$).