Question $2.$ Optimize the following Boolean functions F together with the don$$t care conditions d in

$($a$)$ sum$-$ of$-$ products and $($b$)$ product$-$ of$-$ sums form: $[10$ marks$]$

F$($W$,$ X$,$ Y$,$ Z$)=\backslash$Sigma m $(5,6,11,12),$ d$($W$,$ X$,$ Y$,$ Z$)=\backslash$Sigma m $(0,1,2,9,10,14,15)$

Circuit Inspection

Assume that a combinational circuit with $4$ inputs $\{$A$,$ B$,$ C$,$ D$\}$ and $2$ outputs $\{$F$,$G$\}$ has been defined as follows,

using a hierarchical approach with interim $($i$.$e$.$ first stage output$)$ line values $\{$T$1,$ T$2,$ T$3,$ T$4\}$ defined by:

T$1=$ B$$C T$2=$ A$$B T$3=$ A $+$ T$1$ T$4=$ D xor T$2$

and the final outputs $\{$F$,$ G$\}$ by:

F $=$ T$3+$ T$4$ G $=$ D$+$ T$2$