1. DeMorgan's Theorems Logic Simplification
Source:
Vol. IV - §7 Boolean Algebra

2. From your previous work on Logic Circuits
It can be shown that inverting all Inputs to a gate reverses that gates essential function, for example from
AND to OR
OR to AND
Etc., etc., etc.
and also inverts the output

3. Thus An OR gate w/ all its inputs inverted
(a Negative-OR gate)
behaves the same as a NAND gate
In other words, an AND gate w/ its Output inverted

4. & An AND gate w/ all inputs inverted (a Negative-AND gate)
behaves the same as a NOR gate
IOW’s, an OR gate w/ its Output inverted

5. Question 1 Lets see if we’ve got this down
Take 1 basic gate other than an AND or an OR gate and apply this rule
Using Circuit Maker create a Truth Table that shows this rule is correct…

6. Before you start…
You can use Circuit Maker to create a visual Truth Table all in one simulation
Check out the example on the next slide

7. One Step Truth Tables using Circuit Maker
Using only 2 inputs you can create a visual Truth Table in Circuit Maker
Her you have an AND gate with inverted Outputs (a Negative-AND gate)
In other words a NAND gate

8. Quiz time Now answer the question a couple of slides back
Paste your circuit into a Word doc as question 1

9. So where is this all leading?
There are times when converting a gate from one type to another can be very beneficial
If you have inventory issues or
You need to minimize the number of gates in your circuit to save money

10. So how can you do this?
Over 100 years ago, an English mathematician named DeMorgan came across the Algebra of Boole and asked himself the same question

11. DeMorgan’s work
Resulted in a set of rules that allows one to transform a logic expression from what we today refer to as a
Sum of Products (Minterm) into a
Product of Sums (Maxterm) expression

12. DeMorgan’s Theorems
States the original statement that we looked at earlier “backwards”:
“…that inverting the output of any gate results in the same function as the opposite type gate with inverted inputs”

13. In other words
The AND gate shown w/ inverted output becomes an OR gate w/ inverted inputs

14. For Example
In Boolean Algebra terms one can write:

15. Or…

16. Grouping
A long bar over the term acts as a grouping symbol implying that the output of the expression is inverted
Is the same as

17. Applying DeMorgan’s Theorems
One cannot just break the bar over an expression and rewrite the expression with broken bars
In other words:

18. Quiz: Question 2 Prove it to yourself
Using Circuit Maker & Truth Tables prove that the statement on slide 17
(AB)’ ≠ A’B’
is correct as shown!
Hint: Construct a Truth Table for both scenarios (AB)’ = A’B’ & (AB)’ ≠ A’B’
Prove which one is correct
GO!

19. Instead
One must not only break the bar but change the Boolean Operator as well.
For example

20. The Basis of DeMorgan’s Theorems
It can be though of as breaking the long bar and changing the accompanying operations from * to + or vice versa

21. DeMorgan’s Theorems

22. Subtleties of DMT’s When multiple layers of bars exist
You break them one at a time
Starting with the longest or uppermost bar first
Work your way down until the expression is completely converted

23. Dealing w/ Multiple Layers of Bars
Given (A + (BC)’)’, we have visually

24. Example Cont…
Applying DMT’s, we get mathematically

25. Point of Order
Once there are no groups (expression w/ multiple variables covered by a long bar) that have not been converted
You have succeeded in reducing the expression to its simplest form

26. Example Cont…
Visually the circuit becomes:

27. Warning Will Rogers…
You should never break multiple bars in one step

28. Quiz: Question 3 Using Circuit Maker are correct on slide 24
create a Truth Table(s) that verifies that the last two (2) examples (slide 24 & 27)
are correct on slide 24
and incorrect as done on slide 27
Hint:
create 3 circuits in Circuit Maker
create truth tables from each
conclude

29. Alternate approach
You can start by breaking a shorter grouped or barred section of the expression
But as you’ll see in the next slide it takes more steps
Even though the answer is identical to the result on slide 24

30. Alternate Approach Example

31. Why use Parentheses? In some cases they can be omitted
But in many you would loose the order of operation and the end result would be incorrect
It is important to keep variables formerly group together after breaking together

32. Parentheses, an Example
When correctly applied

33. Parentheses, an Example
When incorrectly applied

34. Applications of DMT's Example 1 Simplify the following Gate Circuit
Lets start:

35. Ex 1 cont… Generate an equivalent Boolean expression
Place sub-expression labels at the output of each gate
Start w/ the first gates you come to and then the next ones, etc., until you have worked your way through the entire circuit

36. Ex 1 cont…
Step 1
Step 2
Step 3

37. Ex 1 cont… mathematically

38. Ex. 1 cont…
The equivalent gate circuit for this much-simplified expression is as follows:

39. Review
DeMorgan's Theorems describe the equivalence between gates with inverted inputs and gates with inverted outputs
Simply put, a NAND gate is equivalent to a Negative-OR gate, and a NOR gate is equivalent to a Negative-AND gate.

40. Review cont…
When "breaking" a complementation bar in a Boolean expression, the operation directly underneath the break (addition or multiplication) reverses, and the broken bar pieces remain over the respective terms

41. Review cont…
It is often easier to approach a problem by breaking the longest (uppermost) bar before breaking any bars under it
You must never attempt to break two bars in one step!

42. Review cont… Complementation bars function as grouping symbols.
Therefore, when a bar is broken, the terms underneath it must remain grouped
Parentheses may be placed around these grouped terms as a help to avoid changing precedence (order of operation)

43. Homework tonight In your textbook, §4, pg 113, do Prob. 62
Convert the Maxterm Boolean expression to its Minterm form:
Show each term as you work through the problem

44. Homework tonight In your textbook, §4, pg 113, do Prob. 63
Convert the Minterm Boolean expression Maxterm to its form:
Show all your work, each step of the way to get full credit

45. References www.allaboutcircuits.com, Vol. IV - §7 Boolean Algebra
Tokheim, Roger L. Digital Electronics Principles and Application, sixth edition,Glencoe/McGraw-Hill,Columbus