1. ENGG 1203 Tutorial Combinational Logic (I) 1 Feb Learning Objectives
Recall Boolean Algebra (SoP/PoS, DeMorgan's Theorem, grouping, redundant)
Simplify logic expressions
News
Lab, TA office hour
Tutorial:
Ack.: HKU ELEC1008, Wikimedia Commons

2. Quick quiz
What is the only set of input conditions that will produce a LOW output for any OR gate?
Any one of the input is LOW
Any one of the input is HIGH
All inputs are LOW
All inputs are HIGH C

3. Quick quiz
What logic level should be applied to the second input of a two-input AND gate if the logic signal at the first input is to be inhibited (prevented) from reaching the output?
A LOW input will keep the output LOW
A LOW input will keep the output HIGH
A HIGH input will keep the output LOW
A HIGH input will keep the output HIGH A

4. Quick quiz
What is the only input combination that will produce a HIGH at the output of a five-input AND gate?
Any one of the input is LOW
Any one of the input is HIGH
All inputs are LOW
All inputs are HIGH D

5. Quick quiz
What is the output expression of the following logic-circuit diagram? D

6. Boolean Algebra A + B = B + A AB = BA A + (B + C) = (A + B) + C
More questions in Appendix
A + B = B + A
AB = BA
A + (B + C) = (A + B) + C
A (BC) = (AB) C
A + BC = (A + B) (A + C)
A (B+C) = AB + AC
A + AB = A
A (A + B) = A
NOT (NOT (A)) = A

7. De Morgan’s Theorem De Morgan's theorem
Bubble pushing via De Morgan's theorem
AND NOT  NOT OR
NOT OR  AND NOT
OR NOT  NOT AND
NOT AND  OR NOT 7

8. Solution 2
Use DeMorgan's Theorem for simplification

9. Boolean Algebra Simplification
Sum of Products
Find out the “1”s
𝑌= 𝐴 ∙𝐵 + 𝐴∙ 𝐵
Better if less “1”
Products of Sum
Find out the “0”s
𝑌= 𝐴 + 𝐵 ∙ 𝐴+𝐵
Better if less “0”

10. Boolean Algebra Using SOP and POS
Find an expression for F and

11. Solution
Sum of Products for F
Product of Sums for F

12. Solution
Sum of Products for
Product of Sums for

13. Representing logic operations
Each function can be represented equivalently in 3 ways:
Truth table – Try every combinations of every input variables
Boolean logic expression – SOP/POS + Simplify the expression
Schematics – Construct from Boolean expressions

14. From logic equations
Boolean expressions  Truth table and logic circuit (AND/OR/NOT) 1 1 1

15. From logic equations
Boolean expressions  Truth table and logic circuit (AND/OR/NOT)

16. From truth tables
Derive the Boolean expression of the output x in terms of the input
Construct the logic circuit using AND gates, OR gates, and INVERTERs.

17. Solution
Extra redundant Terms
1. Construct A/B/C 2. Construct not A/B/C 3. Construct AND gates 4. Construct OR gate 1 2 3 4

18. From truth tables
Truth table  Boolean expressions and logic circuit

19. From schematics
Truth table first?
SOP/POS first?

20. From schematics
XOR

21. Circuit representation of logic equations
Show how can be implemented with one two-input NOR and one two-input NAND gate.
(How to convert ?)
We need to apply De Morgan’s Theorem

22. Circuit representation of digital logic
a) Simplify the circuit shown in the figure using Boolean algebra.
b) Change each NAND gate in the circuit of the figure to a NOR gate, and simplify the circuit using Boolean algebra.

23. Solution (a) Procedure:
1) Obtain the Boolean expression from the circuit 2) Check if we need NAND/NOR gate 3) Simplify the expression by Boolean algebra
Less gate (power and resource)
Shorter “longest path”

24. Solution (b)
First, we convert the circuit

25. Solution (b) Then, we simplify the Boolean expression
(DeMorgan's Theorem)
(Expand)
(Simplify)
(Group, Group)

26. Solution (b)
(Group, Group)
(Simplify)
(Expand)
(Simplify)
(Simplify)

27. Conversion of three representations
Describe the function using Boolean expressions
Draw the truth table and describe the function using sum of product

28. Solution
Approach 1: Boolean simplification  Find TT Approach 2: Construct TT  Find POS
(De Morgan)
(XOR expansion)
(De Morgan)
(De Morgan)
(expansion)
(grouping, expansion)
(cancellation)
POS:

29. Karnaugh map Draw the table  Fill in 0s and 1s  Grouping
Group one/two/four/eight/ sixteen ‘1’(s) only
Use the least number of groups to group all numbers
To group as many numbers as possible in every group

30. Karnaugh map Yellow: Redundant 𝐴 𝐵𝐶 𝐷 +𝐴𝐵𝐶 𝐷 =𝐵𝐶 𝐷
𝐴 𝐵 𝐶𝐷+ 𝐶 𝐷+𝐶 𝐷 + 𝐶 𝐷 =𝐴 𝐵
𝐴𝐵 𝐶 𝐷 +𝐴𝐵 𝐶 𝐷+𝐴 𝐵 𝐶 𝐷 +𝐴 𝐵 𝐶 𝐷=A 𝐶
F(x) = 𝐵𝐶𝐷 + A 𝐶 + 𝐴 𝐵

31. Examples of Karnaugh maps

32. (Appendix) Questions for Boolean algebra

33. Solutions

34. (Appendix) Determining output level from a diagram

35. (Appendix) From logic equations
Draw the circuit diagram to implement the expression
Draw the circuit diagram that implements the expression
using gates having no more than three inputs.

36. (Appendix) Circuit representation of digital logic
Construct the given circuit using NAND gates only
Top down approach: ?
Bottom up approach: ?

37. Solution (a) Top down: Expanding the Boolean expression
By DeMorgan’s Theorems,

38. Solution (b)
Bottom-up: Construct NOT gate, AND gate and OR gate from NAND gate
iii) i)
ii)

39. Solution (b)
Top-down and Bottom-up: Same number of gate, same configuration, different approach
(cancelled)