Question Number 197040 by peter frank last updated on 06/Sep/23
Commented by TheHoneyCat last updated on 08/Sep/23
![a) Q=XY+X^ Z+YZ =(X+X^ )(XY+X^ Z+YZ) =(XY+XYZ)+(X^ Z+X^ YZ) =(X(Y+YZ) )+(X^ (Z+YZ) ) =(XY)+(X^ Z) =XY+X^ Z OR[ AND[X,Y], AND[NOT[X],Z]]](https://www.tinkutara.com/question/Q197135.png)
$$\left.{a}\right)\: \\ $$$${Q}={XY}+\bar {{X}Z}+{YZ} \\ $$$$=\left({X}+\bar {{X}}\right)\left({XY}+\bar {{X}Z}+{YZ}\right) \\ $$$$=\left({XY}+{XYZ}\right)+\left(\bar {{X}Z}+\bar {{X}YZ}\right) \\ $$$$=\left({X}\left({Y}+{YZ}\right)\:\right)+\left(\bar {{X}}\left({Z}+{YZ}\right)\:\right) \\ $$$$=\left({XY}\right)+\left(\bar {{X}Z}\right) \\ $$$$={XY}+\bar {{X}Z} \\ $$$$ \\ $$$$\mathrm{OR}\left[\:\mathrm{AND}\left[{X},{Y}\right],\:\mathrm{AND}\left[\mathrm{NOT}\left[{X}\right],{Z}\right]\right] \\ $$
Commented by TheHoneyCat last updated on 08/Sep/23
![b) Q=XY+XZ +Y(Y+Z)+XY =XY+XZ +Y(Y+Z) =XY+XZ+YY+YZ =XY+XZ+Y+YZ =XY+XZ+Y =XZ+Y OR[Y,AND[X,Z]]](https://www.tinkutara.com/question/Q197136.png)
$$\left.{b}\right) \\ $$$${Q}={XY}+{XZ}\:+{Y}\left({Y}+{Z}\right)+{XY} \\ $$$$={XY}+{XZ}\:+{Y}\left({Y}+{Z}\right) \\ $$$$={XY}+{XZ}+{YY}+{YZ} \\ $$$$={XY}+{XZ}+{Y}+{YZ} \\ $$$$={XY}+{XZ}+{Y} \\ $$$$={XZ}+{Y} \\ $$$$ \\ $$$$\mathrm{OR}\left[{Y},\mathrm{AND}\left[{X},{Z}\right]\right] \\ $$
Commented by TheHoneyCat last updated on 08/Sep/23
![c) let W:=X+Y Q=W(W+Z) =W+WZ =W =X+Y OR[X,Y]](https://www.tinkutara.com/question/Q197137.png)
$$\left.{c}\right) \\ $$$$\mathrm{let}\:{W}:={X}+{Y} \\ $$$${Q}={W}\left({W}+{Z}\right) \\ $$$$={W}+{WZ} \\ $$$$={W} \\ $$$$={X}+{Y} \\ $$$$ \\ $$$$\mathrm{OR}\left[{X},{Y}\right] \\ $$
Commented by TheHoneyCat last updated on 08/Sep/23
![d) let W:= XY Q=W+WZ+WZ^ =W =XY AND[X,Y]](https://www.tinkutara.com/question/Q197138.png)
$$\left.{d}\right) \\ $$$$\mathrm{let}\:{W}:=\:{XY} \\ $$$${Q}={W}+{WZ}+{W}\bar {{Z}} \\ $$$$={W} \\ $$$$={XY} \\ $$$$ \\ $$$$\mathrm{AND}\left[{X},{Y}\right] \\ $$
Commented by peter frank last updated on 10/Sep/23
$$\mathrm{thank}\:\mathrm{you} \\ $$
Commented by TheHoneyCat last updated on 11/Sep/23
you're welcome
Answered by TheHoneyCat last updated on 08/Sep/23
![e) Q=XY^ AND[X,NOT[Y]]](https://www.tinkutara.com/question/Q197139.png)
$$\left.{e}\right) \\ $$$${Q}={X}\bar {{Y}} \\ $$$$ \\ $$$$\mathrm{AND}\left[{X},\mathrm{NOT}\left[{Y}\right]\right] \\ $$