Converse of statements and determining if true

Question

quick statement question here:

Question
The following statements are all true. In each case write out the converse of the statement, and say whether or not the converse is true (give a reason in each case; if it is false, give a counter-example, if it is true, then prove that this is so).

a) Let x ∈ ℝ. If x = 2, then $x^2$ + 1 = 5
b) Let x ∈ ℝ. If x > 4, then $x^2$ > 16
c) Let z ∈ ℂ. If z is real, then $\overline z$ = z

Attempt
Okay so this is my attempt for the converses
a) Let x ∈ ℝ. If $x^2$ + 1 = 5, then x = 2 (which is, I think, true)
b) Let x ∈ ℝ. If $x^2$ > 16, then x > 4 (which isn't true, since, for example $-1^2$ is less then 16)
c) Let z ∈ ℂ. If $\overline z$ = z, then z is real (which I think is true)

So my question is 1) are my attempts valid converses and 2) how would I give a counterexample and go about proving them?

Answer 1

Yes, you have the right converse for all of them.

For a), it isn't true ($x$ could be $-2$).

For b) you don't have a correct counterexample - you need a number such that $x^2>16$, but $x\leq 4$.

For c), you're right that it is true. To see this, write $z=x+iy$, so $\overline z=x-iy$. Then $z-\overline z=2iy$, so $z,\overline z$ are the same if and only if $y=0$.