1. Chapter 2: Boolean Algebra and Logic Functions
CS Digital Logic Design
CS 3402:A.Berrached

2. Boolean Algebra Algebraic structure consisting of: a set of elements B
operations {AND, OR}
Notation: X AND Y X • Y XY
X OR Y X+Y
B contains at least two elements a & b such that a  b
Note: switching algebra is a subset of Boolean algebra where B={0, 1}
Axioms of Boolean Algebra
1. Closure a,b in B,
(i) a + b in B
(ii) a • b in B
2. Identities: 0, 1 in B
(i) a + 0 = a
(ii) a • 1 = a
3. Commutative Laws: a,b in B,
(i) a + b = b + a
(ii) a • b = b • a
4. Associative Laws:
(i) a + (b+c) = (a+b)+c = a+b+c
(ii) a. (b.c) = (a.b).c = a.b.c
5. Distributive Laws:
(i) a + (b • c) = (a + b) • (a + c)
(ii) a • (b + c) = (a • b) + (a • c)
6. Existence of the Complement:
exists a’ unique in B
(i) a + a’ = 1
(ii) a • a’ = 0
a’ is complement of a
CS 3402:A.Berrached

3. Principle of Duality Definition of duality:
a dual of a Boolean expression is derived by replacing AND operations by ORs, OR operations by ANDs, constant 0s by 1s, and 1s by 0s (everything else is left unchanged).
Principle of duality: if a statement is true for an expression, then it is also true for the dual of the expression
Example: find the dual of the following equalities
1) XY+Z = 0
2) a(b+c) = ab + ac
CS 3402:A.Berrached

4. Boolean Functions
A Boolean function consists of an algebraic expression formed with binary variables, the constants 0 and 1, the logic operation symbols, parenthesis, and an equal sign.
Example:
F(X,Y,Z) = X + Y’ Z or F = X + Y’ Z
X, Y and Z are Boolean variables
A literal: The appearance of a variable or its complement in a Boolean expression
A Boolean function can be represented with a truth table
A Boolean function can be represented with a logic circuit diagram composed of logic gates.
CS 3402:A.Berrached

5. From Boolean Expression to Gates
More than one way to map an expression to gates
E.g., Z = A' • B' • (C + D) = (A' • (B' • (C + D))) A Z B T 1 C T 2 D A B Z C D
For each Boolean function, there is only one unique truth table representation
=>Truth table is the unique signature of a Boolean function
CS 3402:A.Berrached

6. Boolean Functions Possible Boolean Functions of Two variables NAND NOR
Description Z
= 1 if X
is 0 or Y
Gates T
ruth T
able
NAND X X 1 Y 1 Z 1 Z Y
Description Z
= 1 if both X
and Y
are 0
Gates T
ruth T
able
NOR X X 1 Y 1 Z 1 Z Y
CS 3402:A.Berrached

7. Basic Logic Functions: NAND, NOR
NAND, NOR gates far outnumber AND, OR in typical designs
easier to construct in the underlying transistor technologies
they are functionally complete
Functionally Complete Operation Set:
A set of logic operations from which any Boolean function can be realized (also called universal operation set)
E.g. {AND, OR, NOT} is functionally complete
The NAND operation is also functionally complete
=> any Boolean function can be realized with one type of gate (the NAND gate).
The NOR operation is also functionally complete
CS 3402:A.Berrached

8. Basic Logic Functions: XOR, XNOR
XOR: X or Y but not both ("inequality", "difference")
XNOR: X and Y are the same ("equality", "coincidence")
CS 3402:A.Berrached

9. Logic Functions: Rationale for Simplification
Logic Minimization: reduce complexity of the gate level implementation
reduce number of literals (gate inputs, circuit inputs)
reduce number of gates
reduce number of levels of gates
fewer inputs implies faster gates in some technologies
fan-ins (number of gate inputs) are limited in some technologies
Fewer circuit inputs implies fewer I/O pins
fewer levels of gates implies reduced signal propagation delays
number of gates (or gate packages) influences manufacturing costs
In general, need to make tradeoff between circuit delay and reduced gate count.
CS 3402:A.Berrached

10. Simplification Using Boolean Algebra
Useful Theorems of Boolean Algebra:
1. Idempotency Theorem
a. X + X = X b. X • X = X
2. Null elements for + and • operators
a. X + 1 = 1 b. X . 0 = 0
3. Involution Theorem
(X’)’ = X
4. Absorption Theorem
a. X + XY = X b. X.(X+Y) = X
5. Simplification Theorem
a. XY + XY’ = X b. (X+Y).(X+Y’) = X
6. Another Simplification Theorem
a. X + X’Y = X + Y b. X.(X’ + Y) = X.Y
CS 3402:A.Berrached

11. DeMorgan's Theorems 7. DeMorgan’s Theorem
a. (X+Y)’ = X’ . Y’ b. (X.Y)’ = X’ + Y’
The complement of the sum is the product of the complements
The complement of the product is the sum of the complements
In general
a. (A+B+….+Z)’ = A’ . B’ . … .Z’
b. (A.B.C….Z)’ = A’ + B’ + ….+Z’
CS 3402:A.Berrached

12. DeMorgan's Theorem (X + Y)' = X' • Y' NOR is equivalent to AND
with inputs complemented
(X • Y)' = X' + Y'
NAND is equivalent to OR
with inputs complemented
DeMorgan’s Law can be used to get the complement of an expression
{F(X1,X2,...,Xn,0,1,+,•)}' = {F(X1',X2',...,Xn',1,0,•,+)}
Example:
F = A B' C' + A' B' C + A B' C + A B C'
F' = (A' + B + C) • (A + B + C') • (A' + B + C') • (A' + B' + C)
CS 3402:A.Berrached

13. Function Representations
Truth Table (Unique representation)
Boolean Expressions
Logic Diagrams
From TO
Boolean Expression ==> Logic Diagram
Logic Diagram ==> Boolean Expression
Boolean Expression ==> Truth Table
Truth Table ==> Boolean Expression
CS 3402:A.Berrached

14. Function Representations
F(X,Y,Z) = X + Y’Z + X’Y’Z + X’Y’Z’
CS 3402:A.Berrached

15. Deriving Boolean Expression from Truth Table
CS 3402:A.Berrached

16. Product and Sum Terms --Definitions
Literal: A boolean variable or its complement
X X’ A B’
Product term: A literal or the logical product (AND) of multiple literals: X XY XYZ X’YZ’ A’BC
Note: X(YZ)'
Sum term: A literal or the logical sum (OR) of multiple literals:
X X’+Y X+Y+Z X’+Y+Z’ A’+B+C
Note: X+(Y+Z)'
CS 3402:A.Berrached

17. SOP & POS -- Definitions
Sum of products (SOP) expression: The logic sum (OR) of multiple product terms:
AB + A’C + B’ + ABC
AB’C + B’D’ + A’CD’
Product of sums (POS) expression: The logic product (AND) of multiple sum terms:
(A+B).( A’+C).B’.( A+B+C)
(A’ + B + C).( C’ + D)
Note:
SOP expressions ==> 2-level AND-OR circuit
POS expressions ==> 2-level OR-AND circuit
CS 3402:A.Berrached

18. Minterms & Maxterms -- Definitions
A Minterm: for an n variable function, a minterm is a product term that contains each of the n variables exactly one time in complemented or uncomplemented form.
Example: if X, Y and Z are the input variables, the minterms are:
X’Y’Z’ X’Y’Z X’YZ’ X’YZ XY’Z’ XY’Z XYZ’ XYZ
A Maxterm: for an n variable function, a maxterm is a sum term that contains each of the n variables exactly one time in complemented or uncomplemented form
Example: if X, Y and Z are the input variables, the maxterms are:
X’+Y’+Z’ X’+Y’+Z X’+Y+Z’ X’+Y+Z X+Y’+Z’ X+Y’+Z X+Y+Z’ X+Y+Z
CS 3402:A.Berrached

19. Minterms For functions of three variables: X, Y, and Z
The bit combination associated with each minterm is the only bit combination for which the minterm is equal to1.
Example: X'Y'Z' = 1 iff X=0, Y=0, and Z=0
Each bit represents one of the variables ( order is important) :
Un-complemented variable ==> 1
Complemented variable ==> 0
CS 3402:A.Berrached

20. Maxterms
The bit combination associated with each Maxterm is the only bit combination for which the Maxterm is equal to 0.
Note: The ith Maxterm is the complement of the ith minterm; That is Mi = mi
CS 3402:A.Berrached

21. Standard (Canonical) forms of an expression
A switching function can be represented by several different, but equivalent, algebraic expressions.
The standard form is a unique algebraic representation of each function.
Standard SOP: sum of minterm form of a switching function
Standard POS: the product of maxterm form ofa switching function
Each switching function has a unique standard SOP and a unique standard POS.
CS 3402:A.Berrached

22. Deriving Boolean Expression from Truth Table
Input Output Minterm
A B C F term designation
A’B’C’ m0
A’B’C m1
A’BC’ m2
A’BC m3
AB’C’ m4
AB’C m5
ABC’ m6
ABC m7
F is 1 iff (A=0 AND B=0 AND C=0) or (A=0 AND B=1 AND C=1)
F is 1 iff (A’=1 AND B’=1 AND C’=1) or (A’=1 AND B=1 AND C=1)
F is 1 iff A’.B’.C’ = 1 OR A’.B.C= 1
F is 1 iff A’B’C’ + A’BC = 1
=> F = A’B’C’ + A’BC => F = m0 + m3 Short-hand notation: F =  m ( 0, 3)
CS 3402:A.Berrached

23. Sum of minterms form
A Boolean function is equal to the sum of minterms for which the output is one.
=> the sum of minterms (also called the standard SOP) form
Example: F =  m ( 0, 3)
CS 3402:A.Berrached

24. Deriving Boolean Expression from Truth Table
Input Output Minterm Maxterm
A B C F term Designation term Designation
A’B’C’ m A + B + C M0
A’B’C m A + B + C’ M1
A’BC’ m A + B’ + C M2
A’BC m A + B’ + C’ M3
AB’C’ m A’ + B + C M4
AB’C m A ‘ + B + C’ M5
ABC’ m A’ + B’ + C M6
ABC m A’ + B’ + C’ M7
F is 0 iff (A+B+C’) = 0 AND (A+B’+C)=0 AND (A’+B+C)=0 AND (A’+B+C’) =0
AND (A’+B’+C) = 0 AND (A’+B’+C’) = 0
=> F = (A+B+C’) . (A+B’+C) . (A’+B+C) AND (A’+B+C’) . (A’+B’+C) . (A’+B’+C’)
=> F = M1.M2.M4.M5.M6.M7 ==> F =  M(1,2,4,5,6,7)
CS 3402:A.Berrached

25. Product of Maxterms
A Boolean function is equal to the product of Maxterms for which the output is 0.
=> the product of Maxterms (also called the standard Product of Sums) form
Example: F =  M(1,2,4,5,6,7)
CS 3402:A.Berrached

26. Examples: Find the truth table for the following switching functions:
F(A,B,C) = ABC’ + AB’C
F(A,B,C) = AB + A’B’ + AC
F(X, Z) = X + Z’
F(A,B,C,D) = A(B’ + CD’) + A’BC’
For each of the above functions, find their Standard SOP and POS.
CS 3402:A.Berrached

27. Getting Standard Forms of a Switching Function
F(A,B,C) = AB + A’B’ + AC
get standard SOP and POS forms of F
Method 1:
1. Derive Truth Table for F
2. Get SOP and POS from truth table
Method 2:
Use Shannon’s Expansion Theorem
CS 3402:A.Berrached

28. Shannon’s Expansion Theorem
a) f(x1,x2,…,xn) = x1.f(1,x2,….,xn) + x1.f(0,x2,…,xn)
b) f(x1,x2,…,xn) = [ x1+ f(0,x2,….,xn)] . [ x1.f(1,x2,…,xn)]
CS 3402:A.Berrached

29. Incompletely Specified Functions
The output for certain input combination is not important (I.e. we don't care about it).
Certain input combinations never occur
Example: Design a circuit that takes as input a BCD digit and outputs a 1 iff the parity of the input is even.
Note: A BCD digit consists of 4 bits
CS 3402:A.Berrached

30. Incompletely Specified Functions
Block Diagram W X F Y Z
CS 3402:A.Berrached

31. Incompletely Specified Functions
Truth Table
F =  m ( 0, 3, 5, 6, 9)+d(10…15)
F =  M(1,2,4,7, 8)+ d(10…15)
CS 3402:A.Berrached