Show that there is an identity element with respect to the operation $+$ and every subset $A$ of $D$ has an inverse with respect to $+$

Question

Let $P_D$ be a power set of $D$.

The operation $+$ is to be regarded as an operation on $P_D$.

Show that there is an identity element with respect to the operation $+$ and every subset $A$ of $D$ has an inverse with respect to $+$.

Let $A * B = A + B.$

$A * e = A + e = A$. Then $e = 0$ which is an identity for $+$.

$A * A' = A + A' = e.$ Then $A' = - A$. Thus, $A' * A = - A + A = 0.$ So, the inverse exists.

Checking to see if that's correct.

Answer 1

It looks like your proof assumes that $*$, which you've defined to be equal to $+$, is a group operation - but you never used any properties of $+$ and your proof would equally well establish that every binary operation on any power set is a group - which is false.

You need to expand $+$ as a symmetric difference, if that is what it is defined as and prove these things.