1. **Problem Statement:** Simplify the given Boolean expressions and find minimal expansions using Karnaugh maps (K-maps). --- **13a) Simplify** $F = AB + BC + B'C$: 1. Use the distributive property: $F = AB + BC + B'C = AB + C(B + B')$ 2. Since $B + B' = 1$, $F = AB + C \cdot 1 = AB + C$ 3. Final simplified form: $F = AB + C$ --- **13b) Simplify** $F = A + A'B$: 1. Apply consensus theorem: $A + A'B = A + B$ 2. Final simplified form: $F = A + B$ --- **13c) Simplify** $F = A'B'C + A'BC + AB'$: 1. Group terms: $A'B'C + A'BC = A'C(B' + B) = A'C \cdot 1 = A'C$ 2. So, $F = A'C + AB'$ 3. Final simplified form: $F = A'C + AB'$ --- **13d) Simplify** $F = AB + (AC)' + AB'C(AB + C)$: 1. Apply De Morgan: $(AC)' = A' + C'$ 2. Expand $AB'C(AB + C) = AB'C \cdot AB + AB'C \cdot C = A B' C B A + A B' C C = 0 + A B' C = A B' C$ 3. So, $F = AB + A' + C' + A B' C$ 4. Since $A' + AB = A' + B$, and $C' + C = 1$, simplify to $F = A' + B + C'$ --- **14a) K-map for** $xy' + x'y'$: 1. Truth table shows $F=1$ when $y=0$ regardless of $x$ 2. Minimal sum: $y'$ --- **14b) K-map for** $xy + xy'$: 1. $F=1$ when $x=1$ regardless of $y$ 2. Minimal sum: $x$ --- **14c) K-map for** $xy + xy' + x'y + x'y'$: 1. Covers all combinations 2. Minimal sum: $1$ --- **15a) K-map for** $x'yz' + x'y'z'$: 1. Group terms: $x' z' (y + y') = x' z'$ 2. Minimal sum: $x' z'$ --- **15b) K-map for** $xyz + xy'z' + x'yz + xy'z'$: 1. Note $xy'z'$ appears twice, count once 2. Group $xyz + x'yz = yz(x + x') = yz$ 3. Group $xy'z' + xy'z' = xy'z'$ 4. Minimal sum: $yz + xy'z'$ --- **15c) K-map for** $xyz' + xy'z + xy'z' + x'yz + x'yz'$: 1. Group $xyz' + xy'z' = x z' (y + y') = x z'$ 2. Group $xy'z + x'yz = y (x' z + x z')$ 3. Group $x'yz'$ 4. Minimal sum: $x z' + y (x' z + x z') + x' y z'$ --- **15d) K-map for** $xyz + xy'z + xy'z' + x'yz + xy'z' + x'y'z'$: 1. Simplify repeated terms 2. Group terms to cover all minterms 3. Minimal sum: $y z + x y' + x' y' z'$ --- **16a) K-map for** $wxyz + wxyz' + wx'yz + wx'yz' + wx'yz$: 1. Simplify repeated terms 2. Group $w y z (x + x') = w y z$ 3. Minimal sum: $w y z$ --- **16b) K-map for** $w x y' z + w x' y z' + w x' y z + w x' y z' + w' x y' z' + w' x' y' z$: 1. Simplify repeated terms 2. Group terms carefully to minimize 3. Minimal sum: $w x y' z + w x' y + w' x y' z' + w' x' y' z$ --- **Summary:** - Boolean simplification uses distributive, consensus, and De Morgan's laws. - K-maps help find minimal sum-of-products by grouping adjacent 1s. - Repeated terms are counted once.