Find the simplest form of given Boolean expressions using algebraic methods. i. A(A+B) + B(B+C) + C(C+A) ii. (A+|B|)(B+C) + (A+B)(C+|A|) iii. (A+B)(AC+A|C|)_+AB +B iv. |A|(A+B) + (B+A)(A+|B|)
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(i))A(A+B)+B(B+C)+C(C+A)⇒A⋅A+A⋅B+B⋅B+B⋅C+C⋅C+C⋅A⇒A+AB+B+BC+C+CA[∵A.A=A,B.B=B,C.C=C]⇒A(1+B)+B(1+C)+C(1+A)⇒A+B+Cii)(A+Bˉ)(B+C)+(A+B)(C+Aˉ)⇒AB+AC+BˉB+BˉC+AC+AAˉ+BC+BAˉ⇒AB+AC+O+BˉC+AC+O+BC+BAˉ⇒AB+BˉC+AC+AC+BC+BAˉ⇒AB+BˉC+AC+BC+BAˉ[AC+AC=AC]⇒AB+BAˉ+AC+BˉC+BC⇒B(A+Aˉ)+AC+C(Bˉ+B)⇒B+AC+C[Aˉ+A=1]⇒B+C(1+A)[C(1+A)=C]⇒B+Ciii)(A+B)(AC+ACˉ)+AB+B⇒(A+B):(A(C+Cˉ))+AB+B⇒(A+B)A+AB+B [∵C+Cˉ=1]⇒A⋅A+B⋅A+AB+B⇒A+BA+AB+B[A.A=A]⇒A(1+B)+B(1+A)⇒A+B[∴A(1+B)=AB(1+A)=B]iv)Aˉ(A+B)+(B+A)⋅(A+Bˉ)⇒AˉA+AˉB+BA+BBˉ+A⋅A+ABˉ⇒0+AˉB+BA+O+A+ABˉ[AˉA=0;BBˉ=0]⇒B(Aˉ+A)+A(1+Bˉ)⇒B+A[∵Aˉ+A=1A(1+Bˉ)=A]