Looking for the solutions of few mathematics questions?

a) ABC' + B C' D' + B C + C' D = B + C' D

Sol:

LHS = (ABC’+BC)+ (BC’D’+C’D)

       =B(C+C’A) +C’ (D+D’B)

       =B(C+A) +C’ (D+B)

       =AB+BC+C’D+C’B

      =AB+C’D+B (C+C’)                //C+C’=1

      =AB+C’D+B

      =B (1+A) +C’D                        //1+A=1

      =B+C’D

      =RHS

b) WY + W' Y Z' + WXZ + W' X Y' = WY + W' X Z' + X' Y Z' + X Y' Z

Sol:

RHS=wy + w’xz’ + x’yz’ + xy’z

        =WY(1+XZ)+ w’xz’(Y+Y’)+ x’yz’ (W+W’)+ xy’z(W+W’)         //Y+Y’=W+W’=1+XZ=1

        = wy + wxyz + w’xyz’ + w’xy’z’ +wx’yz’ + w’x’yz’ + wxy’z + w’xy’z

        =wy + w’xyz’ + w’x’yz’ + wxyz +wxy’z + w’xy’z + w’xy’z’ + wx’yz’

        =wy + wx’yz’ + w’yz’(x + x’) + wxz(y + y’)+w’xy’(z + z’)

        =wy(1 + x’z’) + w’yz’ + wxz + w’xy’

        = WY + W' Y Z' + WXZ + W' X Y'

        =LHS

c) A D' + A' B + C'D + B'C = (A' + B' + C' + D')(A + B + C + D') = AC’+A’B +B‘C + D’

Sol:

RHS=(A' + B' + C' + D')(A + B + C + D')

        =A’A+A’B+A’C+A’D’+B’A+B’B+B’C+B’D’+C’A+C’B+C’C+C’D’+D’A+D’B+D’C+D’D’

        =A’B+A’C+A’D’+B’A+B’C+B’D’+C’A+C’B+C’D’+D’A+D’B+D’C+D’   //AA’=0; D’D’=D’

        = AC’+A’B +B‘C + D’+ A’C (B+B’) + B’A (C+C’) + C’B (A+A’)    //A+A’=1

        = AC’ (1 + B’ + B) + A’B (1 + C’ + C) +B’C(1 + A + A’) + D’    //1+A=1

        = AC’+A’B +B‘C + D’

        =LHS