Show that the complex conjugation function f C rightarrow C

Show that the complex conjugation function f: C rightarrow C (whose rule is f(a + bi) = a - bi) is a bijection.

Solution

Given that f(a+bi) = a-bi

Let us consider two images a-bi and c-id

If a-bi = c-id then a=c and b = d

Hence a+bi = c+id

Thus if f(z1) = f(z2) then z1 =z2 hence one to one

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To prove onto

Let p+qi be any element in C

We can find p and q from the given element and form p-qi

f(p-qi) = p+iq

Thus all complex numbers have preimages hence onto

Thus the complex conjugation function is a bijection.