Hi guys I did two proofs for a logic class that I'm in. Could someone please check these to make sure I'm on the right track?

**For all sets A and B, (A\B)U(AnB)=A.**

Let x E A

if x E B, then x E (AnB)

if x not E B, then x E (A\B)

therefore x E (A\B)U(AnB)

therefore (A\B)U(AnB) = A

**Let S be a subset of a universal set U. The characteristic function fs is the function from U to the set {0,1} defined by**

**fs(x)={1, if x E S**

**{0, if x not E S**

**Let A and B be sets. Show that for all x, fAnB(x) = fA(x) * fB(x)**

Let x E U

Case 1: If x E A ^ x E B, then fsAnB(x) = 1 = 1*1 = fsA(x) * fsB(x)

Case 2: If x E A but x not E B, then fsAnB(x) = 0 = 1*0 = fsA(x) * fsB(x)

Case 3: If x not E A but x E B, then fsAnB(x) = 0 = 0*1 = fsA(x) * fsB(x)

Case 4: If x not E A ^ x not E B, then fsAnB(x) = 0 = 0*0 = fsA(x) * fsB(x)