Question $3[14]$ Use boolean theorems to prove or disprove the following relationships:

$($a$)A+A^{\text{'}}*B=A+B$

$($b$)(A+B)*(A^{\text{'}}+C)=(A*C)+(A^{\text{'}}*B)$

$($c$)(A+B)*(A^{\text{'}}+C)*(B^{\text{'}}+C^{\text{'}})=(A*B^{\text{'}}*C)+(A^{\text{'}}*B*C^{\text{'}})$

$($d$)(A+B^{\text{'}})*B=A*B$

$($e$)(A*B)+(A^{\text{'}}*C)+(B*C)=(A*B)+(A^{\text{'}}*C)$