Proving this logical equivalences

Question

Proving the following logical equivalence:

E:

My method:

Let A be:

Let B be:

=> E is True if

is True because:

The problem is I can't simplify the expression below to T. Not sure if I did something wrong or this can't be proved by this method.

Answer 1

Let $B := p\wedge\neg q$, $C:=q\wedge\neg r$, and $A:=B\wedge(B\vee C)$ .

You wish to show that when $B$ is valued true, that $A$ must be too, and when $B$ is valued false, $A$ must be also.   That is $A\iff B$.

Now $B$ is true exactly when we value $p$ true and $q$ false.   $C$ is true exactly when we value $q$ true and $r$ false; so $C$ and $B$ cannot both be true under the same valuation.

Complete the restricted table $\begin{array}{c:c|c} B & C & A:B\wedge (B\vee C)\\\hline T & F & \\ F & T & \\ F & F & \end{array}$