1. A.Abhari CPS2131 Chapter 3: Gate-Level Minimization Topics in this Chapter: The Map Method Two-Variable Map Three- Variable Map Four/Five variable Map Don’t –Care Conditions Product of Sums Simplification Implementing NAND and NOR circuits

2. A.Abhari CPS2132 The Map Method Truth table of f n is unique but f n can be in many different algebraic forms Simplification by using boolean algebra is often difficult because we don’t know how to proceed Map method or Karnaugh map (K_Map) is simple and straightforward method that produces minimum number of terms.

3. A.Abhari CPS2133 Two-Variable Map A f n variable have 2 n minterms (cells)

4. A.Abhari CPS2134

5. 5 Three- Variable Map Adjacent cells represent minterms that differs by only one variable. Therefore, adjacent cells are identical except for one variable that appears complemented in one cell and uncomplemented in the adjacent cell. Example : F(x,y,z) = ∑ (2,3,4,5)

6. A.Abhari CPS2136 Another example : F(x,y,z) = ∑(3,4,6,7)

7. A.Abhari CPS2137

8. 8

9. 9

10. 10 Four variable Map

11. A.Abhari CPS21311

12. A.Abhari CPS21312

13. A.Abhari CPS21313

14. A.Abhari CPS21314

15. A.Abhari CPS21315

16. A.Abhari CPS21316

17. A.Abhari CPS21317

18. A.Abhari CPS21318

19. A.Abhari CPS21319

20. A.Abhari CPS21320

21. A.Abhari CPS21321

22. A.Abhari CPS21322

23. A.Abhari CPS21323 Multilevel NAND circuits To convert multilevel AND-OR to all NAND: 1.Convert all ANDs with AND-invert 2.Convert all ORs with invert-OR 3.Check the bubbles in diagrams if any of them is not compensated by another small circle along the same line insert an inverter (it is one input NAND gate) or complement the input literal

24. A.Abhari CPS21324

25. A.Abhari CPS21325

26. A.Abhari CPS21326

27. A.Abhari CPS21327

28. A.Abhari CPS21328 Implementing NOR circuits To convert multilevel AND-OR to all NOR: 1.Convert all ORs with OR-invert 2.Convert all ANDs with invert-AND 3.Check the bubbles in diagrams if any of them is not compensated by another small circle along the same line insert an inverter (one input NOR gate) or complement the input literal

29. A.Abhari CPS21329

30. A.Abhari CPS21330