This document discusses Boolean algebra and logic simplification techniques. It covers exclusive-OR and equivalence operations, the consensus theorem, and basic methods for simplifying switching functions using Boolean algebra theorems and properties. These include combining like terms, eliminating terms, eliminating laterals, and adding redundant terms. While some rules of ordinary algebra do not apply to Boolean algebra, equalities involving addition or multiplication of variables are valid.

1. Department of Communication Engineering, NCTU 1
Unit 3 Boolean Algebra
(Continued)
1. Exclusive-OR Operation
2. Consensus Theorem

2. Department of Communication Engineering, NCTU 2
3.1 Multiplying Out and Factoring
Expressions

3. Department of Communication Engineering, NCTU 3
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu
 Distributive laws
 X(Y+Z) = XY+XZ
 X+YZ = (X+Y)(X+Z)
 The third distributive law
 (X+Y)(X' + Z) = XZ+X' Y
 Used for multiplying out

4. Department of Communication Engineering, NCTU 4
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu
 Also used for factoring

5. Department of Communication Engineering, NCTU 5
3.2 Exclusive-OR and Equivalence
Operations

6. Department of Communication Engineering, NCTU 6
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
01 1
11 0
10 1
00 0
C=A ⊕ BA B

7. Department of Communication Engineering, NCTU 7
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.
1 1
+ 0 1
1 0 0
Adder

8. Department of Communication Engineering, NCTU 8
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu
 Theorems applied exclusive-OR
 X⊕0 = X (3-8)
 X⊕1 = X' (3-9)
 X⊕X = 0 (3-10)
 X⊕X' = 1 (3-11)
 X⊕Y = Y⊕X (commutative law) (3-12)
 (X⊕Y)⊕Z = X⊕(Y⊕Z)
= X⊕Y⊕Z (associative law) (3-13)
 X⊕0 = X (3-8)
 X(Y⊕Z) = XY⊕XZ (distributive law) (3-14)
 (X⊕Y)' = X⊕Y' = X'⊕Y = XY+X'Y' (3-15)

9. Department of Communication Engineering, NCTU 9
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
11 1
01 0
00 1
10 0
C=A ≡ BA B

10. Department of Communication Engineering, NCTU 10
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'
01 1
01 0
00 1
10 0
C=(A+B) 'A B
11 1
01 0
00 1
10 0
C=A ≡ BA B
NORXNOR

11. Department of Communication Engineering, NCTU 11
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')

12. Department of Communication Engineering, NCTU 12
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 (by (3-6))
= (A'B' + AB)C' + (A'B + AB')C (by (3-19))
= A'B'C' + ABC' + A'BC + AB'C

13. Department of Communication Engineering, NCTU 13
3.3 The Consensus Theorem

14. Department of Communication Engineering, NCTU 14
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)

15. Department of Communication Engineering, NCTU 15
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
 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
A C D A BD BCD    ABC ACD
A C D A BD
  
   BCD ABC  ACD 

16. Department of Communication Engineering, NCTU 16
3.4 Algebraic Simplification of
Switching Expressions

17. Department of Communication Engineering, NCTU 17
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. a'b + a'bc = a'b
E.g. 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' (3-26)

18. Department of Communication Engineering, NCTU 18
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)

19. Department of Communication Engineering, NCTU 19
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu
 Some of the theorems of Boolean algebra are not true for
ordinary algebra
 If X + Y = X + Z, then Y = Z (not true)
1 + 0 = 1 + 1 but 10
 If XY = XZ, then Y = Z (not true for X=0)
However,
 If Y = Z, then X + Y = X + Z (true)
 If Y = Z, then XY = XZ (true)