Boolean Algebra POS of $F(A,B,C,D)=\bar{B}A(\bar{C}D+\bar{B})+(B\bigoplus C)$

Question

I need to find SOP and POS of: $F(A,B,C,D)=\bar{B}A(\bar{C}D+\bar{B})+(B\bigoplus C)$.

I found the SOP, but I struggle with the POS of this one.

Can anyone give me a clue on how to start?

Thanks, I appreciate that!

Answer 1

$$F(A,B,C,D)=B'A(C'D+B')+(B\oplus C)\\ $$ $$ \Longrightarrow B'A(C'D+B')+B'C+BC'\\ \Longrightarrow B'(C+AB'+AC'D)+BC'\\ $$ Using elimination and a factoring theorem: $$ (X+Y)(X'+Z)=XZ+X'Y $$ we have $$ \Longrightarrow (B+C+A+AC'D)(B'+C')\\ \Longrightarrow (A+B+C)(B'+C')\\ $$ Noting that $X+YY'=X$ $$ \Longrightarrow (A+B+C+DD')(AA'+B'+C'+DD') $$ Also $(X+YZ)=(X+Y)(X+Z)$ $$ \Longrightarrow (A+B+C+D)(A+B+C+D')(A+B'+C'+D')(A+B'+C'+D)(A'+B'+C'+D')(A'+B'+C'+D) $$