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Soit l’expression bool\'eenne suivante : $E(x,y,z,t) = xt + x'z't + xy'zt' + x'y't + x'yzt + xy'z'$ $\grave{A}$ l’aide d’un diagramme de Karnaugh, d\'eterminer une somme r\'eduite de cette expression.

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Transformations of absolute value functions

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Q. Soit l’expression bool\'eenne suivante :

E(x,y,z,t) = xt + x'z't + xy'zt' + x'y't + x'yzt + xy'z'

\grave{A}

l’aide d’un diagramme de Karnaugh, d\'eterminer une somme r\'eduite de cette expression.

Identify Minterms: Identify the minterms from the Boolean expression $E(x,y,z,t) = xt + x'z't + xy'zt' + x'y't + x'yzt + xy'z'$ . $\newline$ - Minterms: $xt$ $(1011)$ , $x'z't$ $(0010)$ , $xy'zt'$ $(1100)$ , $x'y't$ $(0100)$ , $x'yzt$ $xt$ $0$ , $xt$ $1$ $xt$ $2$ .
Construct Karnaugh Map: Construct a $4$ -variable Karnaugh map and place $1$ s in the cells corresponding to the minterms. $\newline$ - K-map layout: $\newline$ - Columns ( $x_t$ ): $00$ , $01$ , $11$ , $10$ $\newline$ - Rows ( $y_z$ ): $00$ , $01$ , $11$ , $10$ $\newline$ - Place $1$ s: $x_t$ $2$ , $x_t$ $3$ , $x_t$ $4$ , $x_t$ $5$ , $x_t$ $6$ , $x_t$ $7$ .
Group $1$ s for Simplification: Group the $1$ s in the Karnaugh map to simplify the expression. $\newline$ - Groups: Group of four covering $(1011, 1001, 1100, 1110)$ , group of two covering $(0010, 0110)$ , and another group of two covering $(0100, 0110)$ .

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