Unit 3 Boolean Algebra (Continued)
1. Exclusive-OR Operation
2. Consensus Theorem
3.1 Multiplying Out and Factoring
Expressions
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
Logic Design Unit 3 Boolean Algebra (Continued) Sau-Hsuan Wu
Also used for factoring
3.2 Exclusive-OR and Equivalence
Operations
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
1 1 0
1 0 1
0 0 0
C=A ⊕ B A B
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 Adder
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)
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
0 0 1
1 0 0
C=A ≡ B A B
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'
0 1 0
0 0 1
1 0 0
C=(A+B) ' A B
0 1 0
0 0 1
1 0 0
C=A ≡ B A B
XNOR NOR
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')
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
3.3 The Consensus Theorem
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)
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
A C D A BD BCD ABC ACD A C D A BD
BCD ABC ACD
3.4 Algebraic Simplification of
Switching Expressions
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‘
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)
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 10
If XY = XZ, then Y = Z (not true for X=0)
However,
If Y = Z, then X + Y = X + Z (true)
