I've got a problem here that I could use help solving. I have simplified it to this point. Using Wolfram Alpha, I know it is still possible. My lecturer did it but I didn't catch all of it. It is frustrating me like mad and I didn't want to come here for just one question but I feel that it is my last resort.
I am stuck on;
(B&&C)||(¬B&&C&&D)||(¬A&&C&&D)||(A&&D)||(A&&B)
For a more eye friendly version;
BC + !BCD + !ACD + AD + AB
I need to extract CD from that but I don't know a rule that can do it for me. Wolfram Alpha is telling me that I can still work out the final answer from here (Which is (A + C)(B + D)).
Please Stack Exchange, you're my only hope.
First note that $$B+B\,'D=(B+BD)+B\,'D=B+(BD+B\,'D)=B+(B+B\,')D=B+D\;.$$ Similarly, $A'C+A=A+C$. Thus, $A'CD+AD=(A'C+A)D=(A+C)D$, and $BC+B\,'CD=(B+B\,'D)C=(B+D)C$, so
$$\begin{align*} BC+B\,'CD+A'CD+AD+AB&=(B+D)C+(A+C)D+AB\\ &=BC+CD+AD+CD+AB\\ &=AB+AD+CB+CD\\ &=A(B+D)+C(B+D)\\ &=(A+C)(B+D)\;. \end{align*}$$