Unit 3 Boolean Algebra
(Continued)
1.
2.
Exclusive-OR Operation
Consensus Theorem
Department of Communication Engineering, NCTU
1
3.1 Multiplying Out and Factoring
Expressions
Department of Communication Engineering, NCTU
2
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

Distributive laws



The third distributive law


X(Y+Z) = XY+XZ
X+YZ = (X+Y)(X+Z)
(X+Y)(X' + Z) = XZ+X' Y
Used for multiplying out
Department of Communication Engineering, NCTU
3
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

Also used for factoring
Department of Communication Engineering, NCTU
4
3.2 Exclusive-OR and Equivalence
Operations
Department of Communication Engineering, NCTU
5
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

Exclusive-OR, (⊕) is defined as follows
0⊕0=0 0⊕1=1


1⊕0=1
1⊕1=0
Exclusive-OR is often abbreviated as XOR
The truth table for X⊕Y is
A B
C=A ⊕ B
0 0
0
0 1
1
1 0
1
1 1
0
Department of Communication Engineering, NCTU
6
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

The logic symbol for X⊕Y

X⊕Y = X’
Y+XY’= (X+Y)(X’
+Y’
)
(X⊕Y) ⊕ Z = X⊕Y ⊕ Z
E.g.
11
Adder
+
01
100


Department of Communication Engineering, NCTU
7
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

Theorems applied exclusive-OR
 X⊕0 = X
 X⊕1 = X'
 X⊕X = 0
 X⊕X' = 1
 X⊕Y = Y⊕X (commutative law)
 (X⊕Y)⊕Z = X⊕(Y⊕Z)
= X⊕Y⊕Z (associative law)
 X⊕0 = X
 X(Y⊕Z) = XY⊕XZ (distributive law)
 (X⊕Y)' = X⊕Y' = X'⊕Y = XY+X'Y'
Department of Communication Engineering, NCTU
(3-8)
(3-9)
(3-10)
(3-11)
(3-12)
(3-13)
(3-8)
(3-14)
(3-15)
8
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

Equivalence operation
0 ≡ 0 = 1 0 ≡ 1=0

1 ≡ 0=0
1 ≡ 1=1
The truth table for X ≡ Y is
A B
C=A ≡ B
0 0
1
0 1
0
1 0
0
1 1
1
Department of Communication Engineering, NCTU
9
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

The logic symbol for X ≡ Y

Equivalence gate is often called exclusive-NOR (XNOR)
(X ≡ Y) = XY + X'Y'

NOR
XNOR
A B
C=A ≡ B
A B
C=(A+B) '
0 0
1
0 0
1
0 1
0
0 1
0
1 0
0
1 0
0
1 1
1
1 1
0
Department of Communication Engineering, NCTU
10
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu


How to simplify an expression that contains XOR or
XNOR
 Substitute X⊕Y with X'Y+XY'
 Substitute X ≡ Y with XY + X'Y'
E.g.
F = (A'B≡C) + (B⊕AC')
= [(A'B)C + (A'B)'C'] + [B'(AC') + B(AC')']
= A'BC + (A+B')C' + AB'C' + B(A' + C)
= B(A'C + A' + C) + C'(A + B' + AB')
= B(A' + C) + C'(A + B')
Department of Communication Engineering, NCTU
11
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

When manipulating expressions that contain several XOR
or XNOR operations:
(XY' + X'Y)' = XY + X'Y'

(3-19)
E.g.
A'⊕B⊕C = [A'B' + (A')'B]⊕C
= (A'B' + AB)C' + (A'B' + AB)'C
= (A'B' + AB)C' + (A'B + AB')C
= A'B'C' + ABC' + A'BC + AB'C
(by (3-6))
(by (3-19))
Department of Communication Engineering, NCTU
12
3.3 The Consensus Theorem
Department of Communication Engineering, NCTU
13
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

XY + X'Z + YZ = XY + X'Z
proof :
XY + X'Z + YZ = XY + X'Z + (X + X')YZ
= (XY + XYZ) + (X'Z + X'YZ)
= XY(1 + Z) + X'Z(1 + Y) = XY + X'Z

The dual form of the consensus theorem is
(X+Y)(X'+Z)(Y+Z) = (X+Y)(X'+Z)
proof :
(X'Y' + XZ' + Y'Z')' = (X'Y' + XZ') '
= (X+Y)(X'+Z)(Y+Z) = (X+Y)(X'+Z)
Department of Communication Engineering, NCTU
14
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu


The final result obtained by application of the consensus
theorem may depend on the order in which terms are
eliminated
E.g A
C D A
BD BCD ABC ACD
A
C
D A
BD BCD ABC ACD


Sometimes, we may add a term using the consensus
theorem, then use the added terms to eliminate other terms
E.g
F = ABCD + B’
CDE + A’
B’+ BCE’
add ACDE
then
F = ABCD + B’
CDE + A’
B’+ BCE’+ ACDE
F = A’
B’+ BCE’+ ACDE
Department of Communication Engineering, NCTU
15
3.4 Algebraic Simplification of
Switching Expressions
Department of Communication Engineering, NCTU
16
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

Basic ways of simplifying switching functions
 Combining terms : use XY + XY' = X
E.g.

abc'd' + abcd' = abd'
Eliminating terms : use X + XY = X or the consensus
theorem
E.g.
E.g.

a'b + a'bc = a'b
a'bc' + bcd + a'bd = a'bc' + bcd
Eliminating laterals : use X + X'Y = X + Y
E.g.
A'B+A'B'C'D'+ABCD'
= A'(B + B'C'D') + ABCD‘
= A'(B + C'D') + ABCD‘
= B(A' + ACD') + A'C'D‘
= B(A' + CD') + A'C'D‘
= A'B + BCD' + A'C'D'
Department of Communication Engineering, NCTU
(3-26)
17
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

Adding redundant terms : add XX‘
, multiply (X+X') etc.
E.g.
WX+XY+X'Z'+WY'Z'
(add WZ' by consensus theorem)
=WX+XY+X'Z'+WY'Z'+WZ' (eliminate WY'Z')
=WX+XY+X'Z'+WZ'
(eliminate WZ')
=WX+XY+X'Z'
(3-27)
Department of Communication Engineering, NCTU
18
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu

Some of the theorems of Boolean algebra are not true for
ordinary algebra

If

If
However,
 If
 If
X + Y = X + Z, then
1 + 0 = 1 + 1 but 10
XY = XZ,
then
Y = Z, then
Y = Z, then
Y = Z (not true)
Y = Z (not true for X=0)
X + Y = X + Z (true)
XY = XZ
(true)
Department of Communication Engineering, NCTU
19