Boolean Algebra

1
Boolean Algebra
2
Basic Concept

• The two possible values in the boolean system are
  zero and one.
• Often we will call these values false and true
  (respectively).
• The symbol represents the logical AND
  operation
• e.g., A B is the result of logically ANDing the
  boolean values A and B.
• When using single letter variable names, this
  text will drop the symbol Therefore, AB also
  represents the logical AND of the variables A and
  B
• (we will also call this the product of A and B ).
• The symbol represents the logical OR
  operation
• e.g., A B is the result of logically ORing the
  boolean values A and B .
• (We will also call this the sum of A and B.)
• Logical complement, negation, or not, is a unary
  operator.
• This text will use the ( ) symbol to denote
  logical negation.
• For example, A denotes the logical NOT of A.

3
Postulates (Assumptions)

• P1 Closure
• P3 Commutativity
• P6 Associativity
• P4 Distribution
• P2 Identity
• P5 Inverse

4
Closure

• The boolean system is closed with respect to a
  binary operator if for every pair of boolean
  values, it produces a boolean result.
• For example, logical AND is closed in the boolean
  system because it accepts only boolean operands
  and produces only boolean results.

5
Commutativity

• A binary operator is said to be commutative
  if AB BA for all possible boolean values A
  and B.

6
Associativity

• A binary operator is said to be associative
  if (A B) C A (B C) for all boolean
  values A, B, and C.

A
B
C
AND
AND

A
B
C
AND
AND
A
B
C
OR
OR

A
B
C
OR
OR
7
Distribution

• Two binary operators and are distributive
  if A (B C) (A B) (A C) for all boolean
  values A, B, and C.
• E.g
• A (B C) AB AC
• A(BC) (AB)(AC)

8
WHYA(BC) (AB)(AC)
AND
OR
AND
OR
0
1
9
WHYA(BC) (AB)(AC)
OR
AND
A
B
C
BC
ABC
AB
A
AC
(AB)(BC)
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
0
1
0
1
1
1
1
1
0
1
0
1
0
1
0
1
1
1
1
1
1
0
1
0
1
1
1
1
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
10
Identity

• A boolean value I is said to be the identity
  element with respect to some binary operator
  if A I A.
• E.g AA A
• A1 A
• A? A
• A0 A
• A1 1

0
1
A
AND
A
0
0
A
A
1
0
0
1
0
1
A
OR
1
1
A
A
A
A
A
11
Inverse

• A boolean value I is said to be the inverse
  element with respect to some binary operator
  if A I B and B?A (i.e., B is the opposite
  value of A in a boolean system).
• E.g Inverse of A A

A
Universal
A
12
Exercise 1
0
1
A
0
1
A
A
A
AND
OR
A
A
1
A
0
A
1
A
0
A
1
A
0
A
1
A
0
A
A
A
0
1
1
0
13
Postulates
14
Prove theorems in boolean algebra using these
postulates.
Any valid expression you can create using the
postulates and theorems of boolean algebra
remains valid if you interchange the operators
and constants appearing in the expression.

• Th1 A A A
• Th2 A A A
• Th3 A 0 A
• Th4 A 1 A
• Th5 A 0 0
• Th6 A 1 1
• Th7 (A B) A B
• Th8 (A B) A B
• Th9 A AB A
• Th10 A (A B) A
• Th11 A AB AB
• Th12 A (A B) AB
• Th13 AB AB A
• Th14 (AB) (A B) A
• Th15 A A 1
• Th16 A A 0

Duality
Duality
DeMorgans Theorems
15
Exercise 2
1
0
1
0
1
0
1
0
16
DeMorgans Theorems
AND


OR
(A B) A B
(A B) A B

17
Th10
A (A B) A
AND

18
Th11
A AB AB
19
Th14
(AB) (A B) A
20
Truth Table
Truth Table for a Function with Three Variables
AND Truth Table
OR Truth Table
Truth Table for a Function with Four Variables
21
Simplify Boolean Algebra

• (ab ab b)
• ( a(bb) b) By P4
• (a b) By P5
• ( (ab) ) By Th8
• ab By definition of not

22
Exercise 3

• b(ac) ab bc c
• ba bc ab bc c By P4
• a(bb) b(c c) c By P4
• a1 b1 c By P5
• a b c By Th4

23
Exercise 4

• ab ab ab
• a(bb) ab By P4
• a1 ab By P5
• a ab
• (a a)(a b) By P4
• 1(a b) By P5
• a b

24
Prove Boolean Algebra
F1 xyz xyz xy
Method 1
F2 xy xz
Method 2
F1 xyz xyz xy xz (yy) xy
xz xy F2 xy xz
25
Canonical (standardized) form

• Any Boolean function can be expressed in a
  canonical form using the dual concepts of
  minterms (sum of minterms) and maxterms (product
  of maxterms).

26
Minterm

• For a boolean function of n variables , a product
  term in which each of the n variables appears
  once (either complemented, or uncomplemented) is
  called a minterm.
• Thus, a minterm is a logical expression of n
  variables consisting of only the logical
  conjunction operator and the complement operator.
• There are 2n minterms of n variables
• a variable in the minterm expression can either
  be in the form of itself or its complement
• two choices per n variables.
• E.g
• 1) A, A
• 2) AB, AB, AB, AB
• 3) ABC, ABC, ABC, ABC, ABC, ABC, ABC,
  ABC

A complemented term, like A' is considered a
binary 0 and a noncomplemented term like a is
considered a binary 1. The number 6 with
(1102), and write the minterm expression as m6
ABC'. So m0 of three variables is A'B'C'(0002)
and m7 would be ABC(1112).
27
Sum of Minterm
Observing that the rows that have an output of 1,
The sum of minterms of F1 is m3 AB The sum of
minterms of F2 is m0 m1 m2 AB AB AB
28
Maxterm

• A maxterm is a logical expression of n variables
  consisting of only the logical disjunction
  operator and the complement operator.
• Maxterms are a dual of the minterm idea. Instead
  of using ANDs and complements, we use ORs and
  complements, and proceed similarly.
• E.g
• 1) A, A
• 2) AB, AB, AB, AB
• 3) ABC, ABC, ABC, ABC, ABC,
  ABC, ABC, ABC
• The complement of a minterm is the respective
  maxterm.
• The number 6 with (1102), and write the maxterm
  expression as M6 ABC(0012). So M0 of three
  variables is ABC(1112) and M7 would be
  ABC(0002).

29
Product of maxterms
The sum of minterms of F1 is m3 AB The product
of maxterms of F1 is M0M1M2 (AB)(AB)(AB)
The product of maxterms of F2 is M3 (AB)