1. **Problem:** Simplify the Boolean expression $xy' + x'y'$ using Karnaugh maps. 2. **Formula and rules:** Karnaugh maps group adjacent 1s to minimize expressions. Adjacent means differing by one variable. 3. **Expression:** $xy' + x'y'$ can be factored as $y'(x + x')$. 4. Since $x + x' = 1$, the expression simplifies to $y' \cdot 1 = y'$. 5. **Answer:** The minimal form is $y'$. 1. **Problem:** Simplify $xyz + xyz' + xy'z$. 2. **Formula and rules:** Use Karnaugh maps to group terms differing by one variable. 3. Group $xyz$ and $xyz'$: $xy(z + z') = xy \cdot 1 = xy$. 4. Expression becomes $xy + xy'z$. 5. No further simplification possible; minimal form is $xy + xy'z$. 1. **Problem:** Simplify $xyz' + xy'z + xy'z' + x'yz + x'yz' + x'y'z$. 2. **Formula and rules:** Use Karnaugh maps to find groups. 3. Group $xyz'$ and $xy'z'$: $x z' (y + y') = x z' \cdot 1 = x z'$. 4. Group $x'yz$ and $x'yz'$: $x' y (z + z') = x' y$. 5. Group $xy'z$ and $x'y'z$: $y' z (x + x') = y' z$. 6. Expression simplifies to $x z' + x' y + y' z$. 1. **Problem:** Simplify $xyz + xyz' + x'yz + x'y'z$. 2. **Formula and rules:** Use Karnaugh maps. 3. Group $xyz$ and $xyz'$: $xy(z + z') = xy$. 4. Expression becomes $xy + x' y z + x' y' z$. 5. Group $x' y z$ and $x' y' z$: $x' z (y + y') = x' z$. 6. Final minimal form: $xy + x' z$. 1. **Problem:** Simplify $xyz' + xy'z + xy'z' + x'yz + x'yz' + x'y'z$. 2. **Formula and rules:** Use Karnaugh maps. 3. Group $xyz'$ and $xy'z'$: $x z' (y + y') = x z'$. 4. Group $xy'z$ and $x'y'z$: $y' z (x + x') = y' z$. 5. Group $x' y z$ and $x' y z'$: $x' y (z + z') = x' y$. 6. Final minimal form: $x z' + y' z + x' y$. 1. **Problem:** Simplify $xyz + xyz' + x'yz + x'y'z$. 2. **Formula and rules:** Use Karnaugh maps. 3. Group $xyz$ and $xyz'$: $xy$. 4. Group $x' y z$ and $x' y' z$: $x' z (y + y') = x' z$. 5. Final minimal form: $xy + x' z$. 1. **Problem:** Simplify $xyz + xyz' + xy'z + xy'z' + x'y'z$. 2. **Formula and rules:** Use Karnaugh maps. 3. Group $xyz$ and $xyz'$: $xy$. 4. Group $xy'z$ and $xy'z'$: $x y'$. 5. Expression becomes $xy + x y' + x' y' z$. 6. $xy + x y' = x(y + y') = x$. 7. Final minimal form: $x + x' y' z$. 1. **Problem:** Simplify $xyzt + xyz't + xyzt' + x'yzt + x'y'zt + x'yzt'$. 2. **Formula and rules:** Use Karnaugh maps. 3. Group $xyzt$ and $xyz't$: $xy t (z + z') = xy t$. 4. Group $xyzt$ and $xyzt'$: $xyz (t + t') = xyz$. 5. Group $x'yzt$ and $x'y'zt$: $x' z t (y + y') = x' z t$. 6. Group $x'yzt$ and $x'yzt'$: $x' y z (t + t') = x' y z$. 7. Expression simplifies to $xy t + xyz + x' z t + x' y z$. 8. Further simplification: $xy t + xyz = xy (t + z)$. 9. Final minimal form: $xy (t + z) + x' z t + x' y z$. 1. **Problem:** Simplify $xyz't + xy'zt + xy'z't + x'yzt + x'yz't + x'yzt' + x'y'z't'$. 2. **Formula and rules:** Use Karnaugh maps. 3. Group $xyz't$ and $xy'z't$: $x t (y' z' + y' z') = x t y' z'$ (same term, no grouping). 4. Group $xy'zt$ and $x'yzt$: no direct grouping. 5. Group $x'yzt$ and $x'yzt'$: $x' y z (t + t') = x' y z$. 6. Group $x' y z' t$ and $x' y z t'$: no direct grouping. 7. Expression is complex; minimal form is $x y z' t + x y' z t + x y' z' t + x' y z + x' y z' t + x' y' z' t'$. 1. **Problem:** Simplify $xy'z't + xy'zt' + xy'z't' + x'y'zt + x'y'z't + x'y'zt' + x'y'z't'$. 2. **Formula and rules:** Use Karnaugh maps. 3. Group $xy'z't$ and $xy'z't'$: $x y' z' (t + t') = x y' z'$. 4. Group $xy' z t'$ alone. 5. Group $x' y' z t$ and $x' y' z t'$: $x' y' z (t + t') = x' y' z$. 6. Group $x' y' z' t$ and $x' y' z' t'$: $x' y' z'$. 7. Expression simplifies to $x y' z' + x y' z t' + x' y' z + x' y' z'$. 8. Group $x' y' (z + z') = x' y'$. 9. Final minimal form: $x y' z' + x y' z t' + x' y'$.