Originally Posted by

**horan** Here you go!

a := ang lives with the emperor

b := bao loves the emperor

c := chang loves the emperor

d := di loves the emperor

e := eng loves the emperor

$\displaystyle a \implies b \equiv \sim a \vee b \equiv T$

$\displaystyle c \implies (a \wedge d) \equiv

\sim c \vee (a \wedge d) \equiv

(a \vee \sim c) \wedge (d \vee \sim c) \equiv T$

$\displaystyle b \wedge d \implies a \wedge e

\equiv \sim (b \wedge d) \vee (a \wedge e) \equiv T$

Want to show that $\displaystyle c \equiv F$

We know that e = F. Then $\displaystyle a \wedge e = F$. But $\displaystyle \sim (b \wedge d) \vee (a \wedge e) \equiv T$, so

we know that $\displaystyle \sim (b \wedge d) \equiv \sim b \vee \sim d \equiv T$. Therefore, either b=F or d=F.

Next, we know that either ~c or a^d is true. We also know that ~a or b is true.

Case 1. Suppose b=F. Then ~a has to be true, and so a is false. Then (a^d) is false, so ~c is true, and so c is false.

Case 2. Suppose d=F. Then ~c has to be true, and so c is false.

Since c is false in both cases, Chang does not love the emperor.