Answer :
To find the kernel of the homomorphism [tex]f: (\mathbb{C}, +) \to (\mathbb{C}, +)[/tex] where [tex]f(x + iy) = iy[/tex], we need to consider what elements in [tex]\mathbb{C}[/tex] map to zero under this function.
The kernel of a homomorphism [tex]f: G \to H[/tex] is the set of all elements [tex]g \in G[/tex] such that [tex]f(g) = 0[/tex] in [tex]H[/tex]. In this case, since [tex]f(x + iy) = iy[/tex] and we're looking for elements mapping to 0, we set [tex]iy = 0[/tex].
Equation:
[tex]iy = 0[/tex]
Since [tex]i \neq 0[/tex], it implies that [tex]y = 0[/tex].Kernel Calculation:
Therefore, the element [tex]x + iy[/tex] must satisfy [tex]y = 0[/tex], which means [tex]x + 0i[/tex].
Thus, the kernel is the set of all real numbers [tex]\mathbb{R}[/tex], as [tex]x + 0i = x[/tex] where [tex]x[/tex] is any real number.
This kernel represents the real component of the complex numbers [tex]\mathbb{C}[/tex].
The correct answer is hence:
(3) [tex]\mathbb{R}[/tex]
