Question

The complement of the Boolean expression X Y’ Z + X’ Y’ Z is

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Question

Find the complement of the following boolean expressions • x’y + xy’ • (AB’ + C)D’ + E • AB(CD’ + C’D) + A’B’(C + D’)(C’ + D) • (X + Y’ + Z)(X’ + Z’)(X + Y)

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1)x′y+xy′=¯xy+x¯y=x⊕y Complement of the expression =¯¯xy+x¯y=¯¯xy⋅¯x¯y=(¯¯x+¯y)⋅¯x+¯¯y) =(x+¯y)⋅(¯x+y)=x⋅¯x+x⋅y+¯y.¯x+¯y.y=x⋅y+¯x⋅¯y=x⊙y Now we can write ¯x⊕y=x⊙y 2)(AB′+C)D′+E=(A¯B+C)¯D+E=A¯B⋅¯D+C¯D+E Complement of the expression=¯A¯B⋅¯D+C¯D+E =¯A¯B.¯D⋅¯C¯D⋅¯E=(¯A¯B+¯¯D)⋅(¯C+¯¯D)⋅¯E =(¯A+¯¯B+D)⋅(¯C+D)⋅¯E =(¯A+B+D)⋅(¯C+D)⋅¯E 3)AB(CD′+C′D)+A′B′(C+D′)(C′+D)=AB(C¯D+¯CD)+¯A.¯B(C+¯D)(¯C+D) =AB(C¯D+¯CD)+¯A.¯B(CD+¯C⋅¯D)=AB(C⊕D)+¯A.¯B(C⊙D) =AB(C⊕D)+¯A.¯B⋅¯(C⊕D) Complement of the expression =¯AB(C⊕D)+¯A.¯B⋅¯(C⊕D) =¯AB(C⊕D)⋅¯¯A.¯B⋅¯(C⊕D) =[¯AB+¯(C⊕D)]⋅[¯¯A.¯B+¯¯(C⊕D)] =[¯A+¯B+¯(C⊕D)]⋅[¯¯A+¯¯B+(C⊕D)] =[¯A+¯B+¯(C⊕D)]⋅[A+B+(C⊕D)] =[¯A+¯B+(C⊙D)]⋅[A+B+(C⊕D)] =[¯A+¯B+(CD+¯C⋅¯D)]⋅[A+B+(C¯D+¯CD)] 4)(X+Y′+Z)(X′+Z′)(X+Y)=(X+¯Y+Z)(¯X+¯Z)(X+Y) =(X+¯Y+Z)(¯XY+X¯Z+Y¯Z) =X¯Z+XY¯Z+X¯Y⋅¯Z+¯XYZ =X¯Z[1+Y+¯Y]+¯XYZ =X¯Z⋅1+¯XYZ =X¯Z+¯XYZ Complement of the expression=¯X¯Z+¯XYZ =(¯X¯Z)⋅(¯¯XYZ)=(¯X+Z).(X+¯Y+¯Z) =¯X⋅¯Y+¯X⋅¯Z+XZ+¯YZ =¯X(¯Y+¯Z)+Z(X+¯Y)

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