Introduction to digital logic, fundamental logic gates, Boolean algebra & simplifications, 3-6 input Kanaugh Maps

1. Digital Logic
CS2052 Computer Architecture
Computer Science & Engineering
University of Moratuwa
Dilum Bandara
Dilum.Bandara@uom.lk

2. Outline
 Logic gates
 Boolean Algebra
 Kanaugh Maps
2

3. Logic Gates
 Every digital device is based on a set of chips
designed to store & process information
 Basic building blocks of these chips are logic
gates
 Implementation of gates can be different
 Different materials & fabrication technologies
 Different operating voltages
 e.g., 5v vs. 0v, 3+v vs. 0.5v
 Today, they go as low as 1.9 - 2.1v
 But their logical behavior is consistent across all
computers
3

4. Common Logic Gates
 NOT/inverter
 AND
4
Truth table
Algebraic function

5. Common Logic Gates (Cont.)
 OR
 NAND
5

6. Common Logic Gates (Cont.)
 NOR
 XOR/EXOR (Exclusive OR)
6

7. Common Logic Gates (Cont.)
 NXOR/EXNOR (Exclusive NOR)
7

8. Exercise
 Implement 3-input AND using 2-inputs ANDs
 Implement a NOT gate using a NAND
8

9. Fundamental Logic Gate
 2-input NAND gate can be used to build any
other gate
9

10. Exercise
 Implement AND using NANDs
 Homework
 Implement following gates using NANDs
 OR
 XOR
10

11. Boolean Algebra
 Boolean variable
 Takes only 2 values – either TRUE (1) or FALSE (0)
 Boolean function
 Mapping from Boolean variables to a Boolean value
 Application
 Can represent complex relationships in a digital circuit
 Boolean algebra
 Deals with binary variables & logic operations
operating on those variables
 Application
 Can simplify a relationship among multiple inputs in a digital
circuit 11

12. Basic Identities of Boolean Algebra
 Basic operations
 AND (.), OR (+), NOT (– or /)
 0 – FALSE
 1 – TRUE
x + 0 = x x + x = x
x · 0 = 0 x . x = x
x + 1 = 1 x + x/ = 1
x · 1 = x x · x / = 0
12

13. Basic Identities (Cont.)
 Commutativity
x + y = y + x
xy = yx
 Associativity
x + ( y + z ) = ( x + y ) + z
x (yz) = (xy) z
 Distributivity
x ( y + z ) = xy + xz
x + yz = ( x + y )( x + z)
13
Textbook has a typo

14. Basic Identities (Cont.)
 DeMorgan’s Theorem
( x + y )/ = x/ y/
( xy )/ = x/ + y/
 Generalized DeMorgan's Theorem
(a + b + … z) / = a / b / … z /
(a.b … z) / = a / + b / + … z /
 Involution
(x /) / = x
14

15. Exercise
 Simplify following functions using Boolean Algebra
 a + ab
 a(a + b)
 a(a / + b)
 a + a/ b
 (a + b.c) /
a + ab = a(1+b)=a
a(a + b) = a.a +ab=a+ab=a(1+b)=a
a + a'b = (a + a')(a + b)=1(a + b) =a+b
a(a' + b) = a.a' +ab=0+ab=ab
(a + b.c)' = a'.b' + a'.c' 15

16. Homework
 Show steps for following simplifications
 (a + b)(a + b') = a
 ab + ab'c = ab + ac
 (a + b)(a + b' + c) = a + bc
 (a(b + z(x + a')))' = a' + b' (z' + x')
 (a(b + c) + a'b)'=b'(a' + c')
 (a + b)(a' + c)(b + c) = (a + b)(a' + c)
 No need to submit answers
16

17. Minterms
 Minterm
 Each combination of variables in a truth table
 As no of digital inputs (Boolean variables)
increase no of minterms increase
 2n minterms for n inputs
 As n increases
 Large truth tables
 Long Boolean functions
 Easier to represent using decimal equivalent of
sum of minterms
 F(a, b, c) = Σ (2, 4, 5, 7) 17

18. Exercise
 Represent all odd numbers between 0 – 7 as a sum of
minterms
 F(a, b, c) = Σ (1, 3, 5, 7)
 Use Boolean Algebra to simplify above function
Σ (1, 3, 5, 7) = a‘b‘c + a‘bc + ab‘c + abc
= a‘c(b‘ + b) + ac (b‘ + b)
= a‘c + ac = c (a‘+ a) = c
18
Sum of Products

19. Karnaugh Map (K-Map)
 Pictorial representation of a truth table
 Map of squares – each square for each minterm
 Adjacent squares change in minterm only by one
variable
19
bc
a
00 01 11 10
0 0 1 3 2
1 4 5 7 6
Decimal equivalent
of minterm

20. Example – 3 Variable K-Map
 Simplify represent of all odd numbers between 0 – 7
 F(a, b, c) = Σ (1, 3, 5, 7)
20
ab
c
00
01
11
10
0 1
1
1
1
1
0
0
0
0
= c

21. Exercise – 3 Variable K-Map
 Simplify following expression using a K-map
21
ab
c
00
01
11
10
0 1
1
0
0
0
1
1
0
0
= a’c’ + a’b’

22. Kanaugh Map – Simplification
 Squares containing 1’s are grouped to get sum-
of-product expression
 Group size must be 2n
 Group must be in rectangular shape
 Each group represents an algebraic term
 Its OK for groups to overlap
 OR of all those terms is the final expression
 Maximum group size  better simplification
 A don’t care (X) can be interpreted as either 0 or 1
when needed, only if it contributes to simplification
 Don’t group Xs together!
22

23. 4 Variable K-Map
23

24. Example – 4 Variable K-Map
24

25. Exercise – 4 Variable K-Map
 What would be a Boolean expression of a circuit
that detects all even numbers between 0 – 15?
 Present as a sum of minterms
25

26. 5 Variable K-Map – Option 1
26
BC
DE
A
1
/0
00 01 11 10
00
01
11
10
0
5
3
1
8124
2
20
11
106
7
14
16
13 9
15
19
18
17
28 24
31
2921
23
3022 26
25
27

27. Example – 5 Variable K-Map
 What would be a Boolean expression of a circuit
to detect all prime numbers between 0 & 31?
27
BC
DE
A
1
/0
00 01 11 10
00
01
11
10
0
5
3
1
8124
2
20
11
106
7
14
16
13 9
15
19
18
17
28 24
31
2921
23
3022 26
25
27

28. 5 Variable K-Map – Option 2
28

29. Example – 5 Variable K-Map – Option 2
29

30. 6 Variable K-Map
30

31. Simplification as Product of Sum
 Some times it is useful to obtain the algebraic
expression as a product of sums
 Can be obtained by finding sum of products for
F/ & converting it to F
 To find F/ Group 0s in a K-map
 F(a, b, c) = Σ (1, 3, 5, 7) = Π (1, 3, 5, 7)
31

32. Exercise – Product of Sum
 Find sum-of-product & product-of-sum
representation of following K-map
32
ab
c
00
01
11
10
0 1
1
x
1
1
0
0
0
1
F(a, b, c) = ab’ + c
F/(a, b, c) = a’c’ + bc’
(F/(a, b, c)) / = (a + c)(b’ + c)
= F(a, b, c)