1. Combinational Logic 1

2. Topics Basics of digital logic Basic functions
Boolean algebra
Gates to implement Boolean functions
Identities and Simplification

3. Binary Logic Binary variables Basic Functions
Can be 0 or 1 (T or F, low or high)
Variables named with single letters in examples
Use words when designing circuits
Basic Functions
AND OR
NOT

4. AND Operator Symbol is dot Or no symbol Truth table ->
Z = X · Y
Or no symbol
Z = XY
Truth table ->
Z is 1 only if
Both X and Y are 1

5. OR Operator Symbol is + Truth table -> Z is 1 if either 1
Not addition
Z = X + Y
Truth table ->
Z is 1 if either 1
Or both!

6. NOT Operator Unary Symbol is bar (or ’) Truth table -> Inversion
Z = X’
Truth table ->
Inversion

7. Gates Circuit diagrams are traditionally used to document circuits
Remember that 0 and 1 are represented by voltages

8. AND Gate
Timing Diagrams

9. OR Gate

10. Inverter

11. More Inputs
Work same way
What’s output?

12. Digital Circuit Representation: Schematic

13. Digital Circuit Representation: Boolean Algebra
For now equations with operators AND, OR, and NOT
Can evaluate terms, then final OR
Alternate representations next

14. Digital Circuit Representation: Truth Table
2n rows
where n # of
variables

15. Functions Can get same truth table with different functions
Usually want simplest function
Fewest gates or using particular types of gates
More on this later

16. Identities Use identities to manipulate functions
On previous slide, I used distributive law
to transform from to

17. Table of Identities

18. Duals
Left and right columns are duals
Replace AND with OR, 0s with 1s

19. Single Variable Identities

20. Commutative
Order independent

21. Associative Independent of order in which we group
So can also be written as
and

22. Distributive
Can substitute arbitrarily large algebraic expressions for the variables

23. DeMorgan’s Theorem Used a lot NOR equals invert AND
NAND equals invert OR

24. Truth Tables for DeMorgan’s

25. Algebraic Manipulation
Consider function

26. Simplify Function
Apply
Apply
Apply

27. Fall 2005
Fewer Gates

28. Consensus Theorem The third term is redundant
Can just drop
Proof in book, but in summary
For third term to be true, Y & Z both 1
Then one of the first two terms must be 1!

29. Complement of a Function
Definition:
1s & 0s swapped in truth table

30. Truth Table of the Complement of a FunctionX Y Z
F = X + Y’Z F’ 1

31. Algebraic Form for Complement
Mechanical way to derive algebraic form for the complement of a function
Take the dual
Recall: Interchange AND & OR, and 1s & 0s
Complement each literal (a literal is a variable complemented or not; e.g. x , x’ , y, y’ each is a literal)

32. Example: Algebraic form for the complement of a function
F = X + Y’Z
To get the complement F’
Take dual of right hand side
X . (Y’ + Z)
Complement each literal: X’ . (Y + Z’)
F’ = X’ . (Y + Z’)

33. Mechanically Go From Truth Table to Function

34. From Truth Table to Function
Consider a truth table
Can implement F
by taking OR of all terms that correspond to rows for which F is 1
“Standard Form” of the function

35. Standard Forms Not necessarily simplest F
But it’s mechanical way to go from truth table to function
Definitions:
Product terms – AND  ĀBZ
Sum terms – OR  X + Ā
This is logical product and sum, not arithmetic

36. Definition: Minterm
Product term in which all variables appear once (complemented or not)
For the variables X, Y and Z example minterms :
X’Y’Z’, X’Y’Z, X’YZ’, …., XYZ

37. Definition: Minterm (continued)
Each minterm represents exactly one combination of the binary variables in a truth table.

38. Truth Tables of Minterms

39. Number of Minterms For n variables, there will be 2n minterms
Minterms are labeled from minterm 0, to minterm 2n-1
m0 , m1 , m2 , … , m2n-2 , m2n-1
For n = 3, we have
m0 , m1 , m2 , m3 , m4 , m5 , m6 , m7

40. Definition: Maxterm
Sum term in which all variables appear once (complemented or not)
For the variables X, Y and Z the maxterms are:
X+Y+Z , X+Y+Z’ …. , X’+Y’+Z’

41. Definition: Maxterms (continued)
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mmmmmmmmmmmmmmmmmmmm
mmmmmmmmmmmmmmmmmmm,m
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,mmmmmmmmmmmmmmmmmmmmmmm
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Maxterm

42. Truth Tables of Maxterms

43. Minterm related to Maxterm
Minterms and maxterms with same subscripts are complements
Example

44. Standard Form of F: Sum of Minterms
OR all of the minterms of truth table for which the function
value is 1
F = m0 + m2 + m5 + m7

45. Complement of F Not surprisingly, just sum of the other minterms
In this case
F’ = m1 + m3 + m4 + m6

46. Product of Maxterms Recall that maxterm is true except for its own row
So M1 is only false for 001

47. Product of Maxterms or F = m0 + m2 + m5 + m7 Remember:
M1 is only false for 001
M3 is only false for 011
M4 is only false for 100
M6 is only false for 110
Can express F as AND of M1, M3, M4, M6 or

48. Recap Working (so far) with AND, OR, and NOT Algebraic identities
Algebraic simplification
Minterms and maxterms
Can now synthesize function from truth table