**Boolean Laws Summary** **Dual:** 1. A + A = A 2. A + A' = 1 3. AA = A 4. AA' = 0 5. AB = BA 6. (AB)C = A(BC) 7. A(B+C) = AB + AC 8. A + AB = A 9. A + A'B = A + B 10. x + xy = x 11. x(x+y) = x 12. (x+y)(x+y') = x 13. (x+y)(x'+z) = xx' + xz + yx' + yz F = AB + BCD → F' = (A' + B')(B' + C' + D') **Using properties of Boolean algebra, reduce the expression to a minimum sum of product form.** x(y + wz) + wxz = xy + xwz + wxz = xy + x(wz + wz) = xy + x(w + z) = x(y + w + z) * Simplify the boolean expression: (A+B+C)(D+E) + (A+B+C)(D+E) = (A+B+C)(D+E+D+E) = A+B+C * Simplify the boolean expression: AB + ABC + ABCD + ABCDE + ABCDEF = AB(1+C+CD+...) = AB * Simplify the following boolean expressions: 1. x(x'+y) = xx' + xy = 0 + xy = xy 2. x+x'y = x(1+y) + x'y = x + x'y = x(1+y) = x 3. (x+y)(x'+y') = xx' + xy' + x'y + yy' = 0 + xy' + x'y + 0 = x **4. xy + x'z + yz** xy + x'z + yz = xy + xy'z + xz + x'yz = xy(1+z) + xz(1+y) = xy + xz **5. (x+y)(x'+z)(y+z)** = (x'y + xz)(y+z) = x'y² + xzy + xyz + xz² = x'y + xz **Draw a circuit that uses only one AND gate and one OR gate.** **1. (A+B+C+D)(A+B+C+E)(A+B+C+F)** = (A+B+C)(DEF) [Image of a circuit diagram with AND and OR gates] **2. WXYZ + W'XYZ + UVXYZ** = XYZ(W+W'+U) = XYZ