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Find the complement of the following boolean expressions

  1. x’y + xy’
  2. (AB’ + C)D’ + E
  3. AB(CD’ + C’D) + A’B’(C + D’)(C’ + D)
  4. (X + Y’ + Z)(X’ + Z’)(X + Y)
in Digital Logic by Boss (10.5k points) | 60 views

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$1)x’y + xy’=\overline{x}y+x\overline{y}=x\oplus y$

Complement of the expression $=\overline{\overline{x}y + x\overline{y}}=\overline{\overline{x}y}\cdot\overline{x\overline{y}}=(\overline{\overline{x}}+\overline{y})\cdot\overline{x}+\overline{\overline{y}})$

$=(x+\overline{y})\cdot(\overline{x}+y)=x\cdot\overline{x}+x\cdot y+\overline{y}.\overline{x}+\overline{y}.y=x\cdot y+\overline{x}\cdot \overline{y}=x\odot y$

Now we can write   ${\color{Magenta} {\overline{x\oplus y}=x\odot y}}$

$2)(AB’ + C)D’ + E=(A\overline{B}+C)\overline{D}+E=A\overline{B}\cdot\overline{D}+C\overline{D}+E$

Complement of the expression$=\overline{A\overline{B}\cdot\overline{D}+C\overline{D}+E}$

$=\overline{A\overline{B}.\overline{D}}\cdot\overline{C\overline{D}}\cdot\overline{E}=(\overline{A\overline{B}}+\overline{\overline{D}})\cdot(\overline{C}+\overline{\overline{D}})\cdot\overline{E}$

$=(\overline{A}+\overline{\overline{B}}+D)\cdot(\overline{C}+D)\cdot\overline{E}$

$=(\overline{A}+B+D)\cdot(\overline{C}+D)\cdot\overline{E}$  

$3)AB(CD’+C’D) + A’B’(C + D’)(C’ + D)=AB(C\overline{D} + \overline{C}D) +\overline{A}.\overline{B}(C + \overline{D})(\overline{C}+ D)$

$=AB(C\overline{D} + \overline{C}D) +\overline{A}.\overline{B}(CD+\overline{C}\cdot\overline{D})=AB(C\oplus D) +\overline{A}.\overline{B}(C\odot D)$

$=AB(C\oplus D) +\overline{A}.\overline{B}\cdot\overline{(C\oplus D)}$

Complement of the expression $=\overline{AB(C\oplus D) +\overline{A}.\overline{B}\cdot\overline{(C\oplus D)}}$

$=\overline{AB(C\oplus D)}\cdot\overline{\overline{A}.\overline{B}\cdot\overline{(C\oplus D)}}$

$=\left [\overline{AB}+\overline{(C\oplus D)} \right ]\cdot\left [\overline{\overline{A}.\overline{B}}+\overline{\overline{(C\oplus D)}}\right]$

$=\left [\overline{A}+\overline{B}+\overline{(C\oplus D)} \right ]\cdot\left [\overline{\overline{A}}+\overline{\overline{B}}+(C\oplus D)\right]$

$=\left [\overline{A}+\overline{B}+\overline{(C\oplus D)} \right ]\cdot\left [A+B+(C\oplus D)\right]$

$=\left [\overline{A}+\overline{B}+(C\odot D) \right ]\cdot\left [A+B+(C\oplus D)\right]$

$=\left [\overline{A}+\overline{B}+(CD+\overline{C}\cdot\overline{D}) \right ]\cdot\left [A+B+(C\overline{D} + \overline{C}D)\right]$

$4)(X + Y’ + Z)(X’ + Z’)(X + Y)=(X + \overline{Y} + Z)(\overline{X} + \overline{Z})(X + Y)$

$=(X + \overline{Y} + Z)(\overline{X}Y+X\overline{Z}+Y\overline{Z})$

$=X\overline{Z}+XY\overline{Z}+X\overline{Y}\cdot\overline{Z}+\overline{X}YZ$

$=X\overline{Z}\left [1+Y+\overline{Y}\right]+\overline{X}YZ$

$=X\overline{Z}\cdot 1+\overline{X}YZ$

$=X\overline{Z}+\overline{X}YZ$

Complement of the expression$=\overline{X\overline{Z}+\overline{X}YZ}$

$=(\overline{X\overline{Z}})\cdot(\overline{\overline{X}YZ})=(\overline{X}+Z).(X+\overline{Y}+\overline{Z})$

$=\overline{X}\cdot\overline{Y}+\overline{X}\cdot\overline{Z}+XZ+\overline{Y}Z$

$=\overline{X}(\overline{Y}+\overline{Z})+Z(X+\overline{Y})$

by Veteran (54.8k points)
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by Boss (35.6k points)

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