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I wanted to ask about this problem.

My book states:

Let the symmetric difference be defined as $A + B=(A\setminus B) \cup (B \setminus A)$.

It proceeds to define a power set as

$P_D= \left\{ A:A \subseteq D\right\}$

It then asks me to show:

  1. Prove that there is an identity element with respect to the operation $+$, which is ______.
  2. Prove every subset $A$ of $D$ has an inverse with respect to $+$, which is ______. Thus, $\left\langle P_D,+\right\rangle$ is a group!

Here's what I got:

  1. Suppose $E$ is the identity set. We want to find $E$ such that $A + E=A$ (we don't need to check $E+A=A$ because $\cup$ is commutative--it will work). Hence $E=\varnothing$ because $(A\setminus \varnothing) \cup (\varnothing \setminus A) = A \cup \varnothing = A$.

  2. Let $I$ denote the inverse set that satisfies $A + I = E$. The solution to $A+I=\varnothing$ is $I=A$. We can verify this: $A+A = ( A\setminus A)\cup(A\setminus A)=\varnothing$.

Obviously, in my answer I did not specifically prove that every subset $A$ of $D$ has an inverse. Since I do not know how to prove this, I cannot prove that "$\left\langle P_D,+\right\rangle$ is a group", as they asked.

How should I fix my proof?

Thank you.

Patrick Shambayati
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  • You need to give an explicit construction for $I$. It looks to me like you know what it is, so I'm confused about where you are having difficulty? – Eric Stucky Aug 19 '14 at 16:40
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    Oh... wow. I just realized that since $I=A$ then $I\subseteq D$, so I actually did prove it for $P_D$. This seems very clear to me now. Thanks! – Patrick Shambayati Aug 19 '14 at 16:43

1 Answers1

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Since you already solve the problem, this answer should play as a comment but I put it here for proper format.

$P_D$ can be identified with the set of functions $$f:D\to \Bbb Z/2,$$ from $D$ to the integers mod $2$, where each subset $A$ is identified to its characteristic function $1_A$ such that $$1_A(a)=1\iff a\in A.$$ Then the operation $A+B$ translates to $1_A+1_B$ in normal sense.

It follows easily that

  1. the identity is the zero function, i.e. the characteristic function of the empty set, and
  2. the inverse of any function is itself.
Quang Hoang
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