Group homomorphism with non-trivial kernel

Question

Am I correct in saying that this group homomorphism below has a non-trivial kernel as it is not injective?

$ φ : (\mathbb R, +) → (\mathbb Z, +) : x → b$, where $b$ is the largest integer which is less or equal to $x$.

Answer 1

This is not a homomorphism at all. According to how you defined it $\varphi(1.5+1.5)=\varphi(3)=3$. However, $\varphi(1.5)+\varphi(1.5)=1+1=2$.

Anyway, a group homomorphism is injective if and only if its kernel is trivial.