Digital Design Session 8

Digital Design – © 2020 Mouli Sankaran Email: mouli.sankaran@yahoo.com 2
Session 8: Focus
 Rules of Boolean Algebra
◦ Rules 7 to 12
 Theorems of Boolean Algebra
◦ Consensus Theorem
◦ DeMargon’s Theorems

Rules
of
Boolean Algebra

Rules of Boolean Algebra
 A, B or C can represent a single variable or a combination of
variables
 These 12 rules are useful in simplifying Boolean expressions

Rule 7
 A . A = A (Idempotent or Identity Law)
 Valid for any number of the same input:
 A . A . A ….. A = A

Rule 8
 A . = 0A

Rule 9
 (Involution law)

Rule 10
 A + AB = A (Absorption or Redundancy Law)
 A + AB = A (1 + B) : Factoring (Distributive law)
 = A . 1 : By Rule 2, (1 + B) = 1
 = A : By Rule 4, A . 1 = A
 A + AB = A

Rule 11
 A + B = A + BA

Rule 12
 (A + B) . (A + C) = A + BC

Consensus Theorem

Consensus Theorem (in SOP)
 It states that in the SOP form: (Sum Of Products)
 Let us verify it with Truth table:
14

Consensus Theorem … (SOP form)
 It states that in the SOP form: (Sum Of Products)
 If in a given Boolean expression we can identify two
terms with one having a variable (X.Y) and the other
having its complement ( . Z)
 Then the term that is formed by the product of the
remaining variables (Y.Z) in the two terms in the case
of a SOP expression will be redundant
15
X

Consensus Theorem … (POS form)
 Consensus theorem states that in the POS form: (Product Of
Sums)
 If in a given Boolean expression we can identify two
terms with one having a variable (X + Y) and the other
having its complement ( + Z)
 Then the term that is formed by the product of the
remaining variables (Y + Z) in the two terms in the
case of a POS expression will be redundant
16
X

Quiz 1: Using Consensus Theorem
 Simplify the following Boolean Expression Using Consensus
Theorem
17
 The redundant terms are: BCD
 and ACD
 The simplified expression is:

DeMorgon’s Theorems

Augustus De Morgan
19
Augustus De Morgan,
(born June 27, 1806, Madura, India)
English mathematician and logician
whose major contributions to the
study of logic include the formulation
of De Morgan's laws
DeMorgan, a mathematician who knew Boole,
proposed two important theorems of Boolean Algebra

DeMorgon’s Theorem 1
 Theorem 1:
 The complement of a sum of variables is equal to the product
of the complements of the variables
 It can also be stated as:
 The complement of two or more ORed variables is
equivalent to the AND of the complements of the
individual variables.
 Valid for any number of variables:
20

21

Proof for DeMorgan’s Theorem 1
 Let us assume all the variables (X1 to Xn) are in a
logic 0 state
 Therefore, LHS = RHS

Proof for DeMorgan’s Theorem 1 ..contd
 Let us assume that any one of the n variables, say X1
is in a logic 1 (HIGH) state
 Therefore again, LHS = RHS
 The same holds good when more than one or all
variables in the logic 1 state. Therefore Theorem 1
stands proved.

 Theorem 2:
 The complement of a product of variables is equal to the sum
of the complements of the variables
 It can also be stated as:
 The complement of two or more ANDed variables is
equivalent to the OR of the complements of the
individual variables.
 Valid for any number of variables:
24

25
Home work:
Proof for Theorem 2

Session 8: Summary
 Rules of Boolean Algebra
◦ Rules 7 to 12
 Theorems of Boolean Algebra
◦ Consensus Theorem
◦ DeMargon’s Theorems

References
Ref 1 Ref 2

Digital Design Session 8

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Digital Design Session 8