bijection from $\mathbb{Z}\times\mathbb{Z}_{2}\rightarrow\mathbb{Z}$

Question

I wish to prove there does not exist a bijection relation $$F:\mathbb{Z}\times\mathbb{Z}_{2}\rightarrow\mathbb{Z}$$ Im thinking that there is more elements in $\mathbb{Z}\times\mathbb{Z}_{2}$ but both sets have an infinite number of elements so Im not too sure how that works. But I think another way of saying this is that for each integer we can assign three elements in $\mathbb{Z}\times\mathbb{Z}_{2}$ and hence the relation cant be injective. I am unsure how to formalize this and if this is even enough to show there cannot be a bijection.