$(10)$ Show that $A{B}^{\text{'}}C+AC=AC$

$(10)$ Show that $(A+C^{\text{'}})(AB+BC)=AB$

$(15)$ Use DeMorgan's Theorem to simplify $((A^{\text{'}}+B)(AB+C^{\text{'}})),$

$(15)$ Write a truth table for the following inputs $(A,B,C)$ and output $($Z$).$

${?}^{**}A==1$ if month number is odd $($Jan$.,$ Mar., May, etc.$)$

${?}^{**}B==1$ if month number $7($July$,$ Aug., Sept., etc.$)$

${?}^{**}C==1$ if month is July

${?}^{**}Z==1$ if month has $31$ days

$(25)$ Find the equation for $Z$ using any method.

$\backslash$table$[[$A$,B,$C$,$Z$],[0,0,0,0],[0,0,1,1],[0,1,0,1],[0,1,1,0],[1,0,0,0],[1,0,1,0],[1,1,0,1],[1,1,1,0]]$

$(25)$ Use K$-$Maps to find the equation for Z$.$

$\backslash$table$[[$A$,$C$,,$Z$],[0,0,0,1],[0,0,1,1],[0,1,0,0],[0,1,1,1],[1,0,0,1],[1,0,1,1],[1,1,0,0]]$