Given the function $,$ answer the following questions: $(14$ points$)$

a$)$ Use properties of Boolean algebra to show $.$

b$)$ Find the canonical sum of product $($SOP$).$

c$)$ Find the canonical product of sum $($POS$).$

d$)$ Find the min$-$term expression.

e$)$ Find the max$-$term expression.

f$)$ Find the minimum sum of product using K$-$MAP.

g$)$ Find the minimum product of sum using K$-$MAP.

AAACEXicbZC$7$SgNBFIZnvcZ$4$i$1$raDAYhooRdUbQRgjaWEUwiJCGcOZlNBmcvzMwKYdlXsPFVbCwUsbWz$822$cJFt$4+2$Hg$4$z$/$ncOb$8$LJZCG$9$f$9$dGZm$5+$YXFgtLxeWV$1$bX$10$sZmU$0$eJQt$7$ASEbqhoHmUoS$8$YYSR$/$CZWHAImeYvdXozrrTuutIjCazOKeTeAQSh$8$gWCs$1$StV$/$AocsAPco$2$e$0$w$0$ClkDGk$+1$NmGeaEWa$9$UdqvuRPQveDmUSa$56$r$/$TR$6$UeYBDw$0$KEHrtufGppuCMgIlz$4$qdRPMY$8$BYGvG$0$xhIDrbjq$5$KKO$71$ulTP$1$L$2$hYZO$3$O$8$TKQRajwJmOwMwQ$/27$Njb$/$q$7$UT$4592$UxHGieEhThf$5$iaQmouN$4$aF$8$ojkaOLAAqYf$9$KcQgK$0$NgQizYE$7/$fJf$6$F$5$WPWOq$+7$VUbl$2$nsdRINtkh$1$SIR$05$IjVySOmkQJPfkkTyTF$+$fBeXJenbdp$64$yTz$2$yRH$3$LevwBx$0$ZuQ

f $($a$,$ b$,$ c$)=$abc $+$bc $+$c AAACAnicbVDLSgMxFM$3$UV$62$vUVfiJliECqXMiKIboejGZQX$7$gLaUO$2$mmDc$1$MhiQjlKG$48$VfcuFDErV$/$hzr$8$x$085$CWw$8$EDuecm$+$QeL$+$JMacf$5$tnJLyyura$/$n$1$wsbm$1$vaOvbvXUCKWhNaJ$4$EK$2$PFCUs$5$DWNdOctiJJIfA$4$bXqjm$9$RvPlCpmAjv$9$Tii$3$QAGIfMZAW$2$knn$3$gl$6$DslckJvsIdYZLpRQl$4$ZNKzi$07$FmQIvEjcjRZSh$1$rO$/$On$1$B$4$oCGmnBQqu$06$ke$4$mIDUjnE$4$KnVjRCMgIBrRtaAgBVd$1$kusIEHxulj$30$hzQk$1$nqq$/$JxIIlBoHnkkGoIdq$3$kvF$/7$x$2$rP$3$LbsLCKNY$0$JLOH$/$JhjLXDaB$+4$zSYnmY$0$OASGb$+$iskQJBBtWiuYEtz$5$lRdJ$47$Tinlecu$7$Ni$9$TqrI$48$O$0$REqIRddoCq$6$RTVURwQ$9$omf$0$it$6$sJ$+$vFerc$+$ZtGclc$3$soz$+$wPn$8$AeE$2$WOQ$==$

f $($a$,$ b$,$ c$)=$ abc