$1-$ show that They are equal by using the Boolean Algebra Rules

Draw $"($you can use the software$)"$ the circuit that represents these fictions

$2-$ The RHS$-$ LHS Using AND $-$ OR$-$ NOT gate

a$)$ Prove $x^{\text{'}}{Y}^{\text{'}}+x^{\text{'}}Y++xY=x^{\text{'}}+Y$

b$)$ Prove $A^{\text{'}}B+B^{\text{'}}{C}^{\text{'}}+AB+B^{\text{'}}C=1$

c$)$ Prove $,Y+x^{\text{'}}Z+x{Y}^{\text{'}}=x+Y+x$

d$),(A+B+C{)}^{\text{'}}(ABC{)}^{\text{'}}=A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$

e$),$ABC$+AC=A(B+C)$

f$),A^{\text{'}}{B}^{\text{'}}D+A^{\text{'}}{C}^{\text{'}}D+BD=(A^{\text{'}}+B)D$