Show by an explicit example that the function $h$ given is not injective and is surjective

Question

I'm not to sure how to do this:

Show by an explicit example that the function $h$ given is not injective and is surjective.

$$h(m,n) = m+2n$$ where $m,n \in \mathbb Z$.

I thought maybe if I did $h(5,-3)$ it might be the solution, but for injective I would get the same answer on both side meaning "it is" injective? Can anyone help me?

Answer 1

I assume that $h:\mathbb Z\times\mathbb Z \to \mathbb Z$. Write out the definition of injective, For all $(m,n)\in \mathbb Z\times\mathbb Z$ if $h(m_1,n_1)=h(m_2,n_2)$ then $(m_1,n_1)=(m_2,n_2)$. This is false for $h$ as you said, so give a counterexample. One would be, $h(2,0)=h(0,1)=2$ but $(2,0)\not=(0,1)$. Use the definition of surjective to prove the next statement, for any $y\in \mathbb Z$ there exists $(m,n)\in \mathbb Z\times\mathbb Z$ so that $h(m,n)=y$. An example would be for any $y \in \mathbb Z$ let $(m,n)=(y,0)$. Then $h(y,0)=y+2(0)=y$ so $h$ is surjective.