The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme. The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.

1. Logic equation simplification. Digital Logic and Software Applications © University of Wales Newport 2009 This work is licensed under a Creative Commons Attribution 2.0 License .

13. 1 1 1 1 1 0 1 0 1 0 0 1 0 0 Y 1 0 B A Y B A Logic Equation Simplification

14. 1 1 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 Y 1 0 B A Y B A STEP 1 Logic Equation Simplification

15. STEP 2 Logic Equation Simplification A B Y B A 0 1 Y 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1

16. STEP 3 A always 1 so A B 1 and 0 so no B Expression A 1 and 0 so no A B always 0 so not B Expression Logic Equation Simplification A B Y B A 0 1 Y 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1

17. STEP 4 Complete expression: Logic Equation Simplification A B Y B A 0 1 Y 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1

20. Example A function F has the truth table shown below. Determine the simplest Boolean Expression for the function. Logic Equation Simplification A B C F A 0 0 1 1 0 0 0 1 C B 0 1 1 0 F 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1

21. 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 F 0 1 1 0 C B 1 0 0 0 1 1 0 0 A F C B A Logic Equation Simplification

22. 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 F 0 1 1 0 C B 1 0 0 0 1 1 0 0 A F C B A Logic Equation Simplification

23. 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 F 0 1 1 0 C B 1 0 0 0 1 1 0 0 A F C B A A always 1 so A B 1 and 0 so no B C 1 and 0 so no C Expression A 1 and 0 so no A B always 1 so B C always 1 so C Expression A 1 and 0 so no A B always 0 so not B C always 0 so not C Expression

24. 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 F 0 1 1 0 C B 1 0 0 0 1 1 0 0 A F C B A Complete expression Logic Equation Simplification

25. Example Three judges A, B and C vote: 1 guilty and 0 not guilty. Design a logic circuit using NAND only which will allow a majority decision (F) to be found. e.g. A = 1, B = 0, C = 0 gives an output of 0 (not guilty) Logic Equation Simplification A B C F A 0 0 1 1 0 0 0 C B 0 1 1 0 F 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1

26. 4-input Karnaugh Map This has 16 entries on the Truth Table and so the Karnaugh Map has 16 squares Logic Equation Simplification A B C D Y A 0 0 1 1 0 0 0 0 C D B 0 1 1 0 Y 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1

27. 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 Y 0 1 1 0 C D B 0 0 0 0 1 1 0 0 A Y D C B A Note there is one additional rule for grouping 1’s on this map and larger maps: Rule: 1’s may be grouped between the top row and the bottom row. Logic Equation Simplification

29. 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 Y 0 1 1 0 C D B 0 0 0 0 0 1 1 0 0 A Y D C B A Logic Equation Simplification

30. 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 Y 0 1 1 0 C D B 0 0 0 0 0 1 1 0 0 A Y D C B A Now form groups Logic Equation Simplification

31. 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 Y 0 1 1 0 C D B 0 0 0 0 0 1 1 0 0 A Y D C B A Identify groups A always 1 so A B 1 and 0 so no B C always 1 so C D 1 and 0 so no D Expression A always 1 so A B 1 and 0 so no B C 1 and 0 so no C D always 1 so D Expression A always 1 so A B always 1 so B C 1 and 0 so no C D 1 and 0 so no D Expression A 1 and 0 so no A B always 1 so B C always 1 so C D always 1 so D Expression

32. 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 Y 0 1 1 0 C D B 0 0 0 0 0 1 1 0 0 A Y D C B A Boolean Expression Logic Equation Simplification

34. Logic Equation Simplification 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 G 0 1 1 0 C D B 0 0 0 0 1 1 0 0 A E L G D C B A

35. Expression G Expression L Expression E Logic Equation Simplification A 0 0 1 1 C D B 0 1 1 0 L 0 0 0 1 1 1 1 0 A 0 0 1 1 C D B 0 1 1 0 E 0 0 0 1 1 1 1 0

36. This resource was created by the University of Wales Newport and released as an open educational resource through the Open Engineering Resources project of the HE Academy Engineering Subject Centre. The Open Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme. © 2009 University of Wales Newport This work is licensed under a Creative Commons Attribution 2.0 License . The JISC logo is licensed under the terms of the Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 UK: England & Wales Licence. All reproductions must comply with the terms of that licence. The HEA logo is owned by the Higher Education Academy Limited may be freely distributed and copied for educational purposes only, provided that appropriate acknowledgement is given to the Higher Education Academy as the copyright holder and original publisher. The name and logo of University of Wales Newport is a trade mark and all rights in it are reserved. The name and logo should not be reproduced without the express authorisation of the University. Logic Equation Simplification