I have been asked to find the converse of the following statement:
“For all $x \in \Bbb Z$, if $x \ne 0$, then there is exactly one $y$ such that $xy = x$.”
First I have converted the statement to logical notation and am left with:
$$\forall \, x \in \Bbb Z \quad \exists \, y \in \Bbb Z\, , \; (xy=x) \Rightarrow (x \ne 0)$$
I guess this is the part I am confused about. It seems that if $P \Rightarrow Q$ is true, then the converse, or $Q \Rightarrow P$ is not necessarily true. When crafting the converse statement, does the result still need to be true?
If I don't do this, and literally just switch $P$ and $Q$, I am left with:
$$\forall\, x \in \Bbb Z \quad \exists\, y \in \Bbb Z\, ,\; (x \ne 0) \Rightarrow (xy=x)$$
Which is a false statement.
Can anybody offer any insight/advice?