Question

Let C be the set of complex numbers. The mapping f : C → R given by f (z) = |z|, ∀ z ∈ C, is

• A. one-one onto
• B. many-one onto
• C. many-one into
• D. one-one into

Answer 1

The correct option is C many-one into

Let C be the set of complex numbers.
The mapping f : C → R is given by f (z) = |z|, ∀ z ∈ C.
Let us consider a = 1 + i and b = 1 - i.
Therefore, f(a) = $\sqrt{{\left(1\right)}^2+{\left(1\right)}^2}=\sqrt{2}$
and f(b) = $\sqrt{{\left(1\right)}^2+{\left(-1\right)}^2}=\sqrt{2}$
Here, f(a) = f(b) but a $\ne$ b.
Therefore, f is many-one.
Since, |z| is always non-negative. Therefore, all the negative numbers in co-domain of f are not in the range of f.
Hence, f is into.