logic design - chapter 3: boolean algebra
DESCRIPTION
- Describing Logic Circuits Algebraically. - rules of evaluating a Boolean expression. - Boolean Theorems. - DeMorgan's Theorem. - Universality of NAND and NOR Gates. - Alternate Logic Gate Representations. - Minterms and Maxterms. - STANDARD FORMS.TRANSCRIPT
CHAPTER 3
Boolean Algebra
Contents
Describing Logic Circuits Algebraically rules of evaluating a Boolean expression Boolean Theorems DeMorgan's Theorem Universality of NAND and NOR Gates Alternate Logic Gate Representations Minterms and Maxterms STANDARD FORMS
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Describing Logic Circuits Algebraically OR gate, AND gate, and NOT circuit are
the basic building blocks of digital systems
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Circuits containing Inverters
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Evaluating Logic Circuit Outputs Let A=0, B=0, C=1, D=1, E=1
X = [D+ ((A+B)C)'] • E
= [1 + ((0+0)1 )'] • 1
= [1 + (0•1)'] • 1
= [1+ 0'] •1
= [1+ 1 ] • 1
= 1
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Rules of evaluating a Boolean expression First, perform all inversions of single
terms; that is, 0 = 1 or 1 = 0. Then perform all operations within
parentheses. Perform an AND operation before an OR
operation unless parentheses indicate otherwise.
If an expression has a bar over it, perform the operations of the expression first and then invert the result.
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Determining Output Level from a Diagram
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Boolean Theorems
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Multivariable Theorems(9) x + y = y + x (Commutative law)
(10) x • y = y • x (Commutative law)
(11) x+ (y+z) = (x+y) +z = x+y+z (Associative law)
(12) x (yz) = (xy) z = xyz (Associative law)
(13.a) x (y+z) = xy + xz (Distributive law)
(13.b) x + yz = (x + y) (x + z) (Distributive law)
(13.c) (w+x)(y+z) = wy + xy + wz + xz
(14) x + xy = x (Absorption) [proof]
(15) x + x'y = x + y
(16) (x +y)(x + z) = x +yz
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Proof of (14, 15, 16)x + xy = x (1+y)
= x • 1 [using theorem (6)]
= x [using theorem (2)]
x + x’y = ( x + x’) (x + y) [theorem 13b]
= 1 (x +y)
= (x + y)
(x +y)(x + z) =xx + xz + yx + yz
= x + xz + yx + yz
= x (1+z+y) +yz
= x . 1 + yz
= x + yz
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DeMorgan's Theorem (18) (x+y)' = x' • y' (19) (x•y)' = x' + y' Example
X = [(A'+C) • (B+D')]'
= (A'+C)' + (B+D')'
= (AC') + (B'D)
= AC' + B'D
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Three Variables DeMorgan's Theorem (20) (x+y+z)' = x' • y' • z' (21) (xyz)' = x' + y' + z‘
EXAMPLE: Apply DeMorgan’s theorems to each of the
following expressions: (a) (b) (c)
D)CBA( DEFABC
EFDCBA 12
Universality of NAND Gates
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Universality of NOR Gates
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Alternate Logic Gate Representations
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Minterms and Maxtermsx
y
z
Minterms
Maxterms
Term
Designation
Term
Designation
0
0
0
x' y’ z'
m0
x+y+z
+y
-t- Z
M0
0
0
1
x' y' z
M1
x+y+z’
+y
+ 2
, t
M1
0
1
0
x' y z’ m2 x+y’+z
+ y'
+
z
M2
0
1
1
x' y z
m3 x+y’+z’
+y'
+
2'
M3
1
0
0
x y' z’
m4 x'+y+z
'+y
4-
2
M4
1
0
1
x y' z
m5 x
'+y+z’ M5
1
1
0
x y z’
m6 x
'+y’+z
M6
1
1
1
x y z
m7 x
'+y’+z’
M7
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Canonical FORMS There are two types of canonical forms:
the sum of minterms The product of maxterms
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Sum of minterms f1 = x'y'z + xy'z' + xyz = m1 + m4 +m7
f2 = x'yz + xy'z + xyz’ + xyz = m3 + m5+ m6 + m7
x y Z f1 f2 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1
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Product of maxterms The complement of f1 is read by forming a
minterm for each combination that produces a 0 as:
f1’=x’y’z’ + x’yz’ + x’yz + xy’z + xyz’
f1 = (x + y + z)(x + y' + z)(x + y' + z' )(x’+ y + z)(x’ + y' + z)
= Mo M2 M3 M5 M6
Similarly f2 = ?
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Example: Sum of Minterms Express the Boolean function F = A + B'C
in a sum of minterms. F=A+B'C = ABC + ABC' + AB'C + AB'C' + AB'C + A'B'C
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Example: Product of Maxterms Express the Boolean function F =xy' + yz in
a product of maxterm form. F = xy' + yz = (xy' + y)(xy' + z) = (x + y)(y' + y)(x + z)(y' + z)
= (x + y)(x + z)(y' + z) = (x + y + zz')(x + yy' + z)(xx' + y' + z) = (x + y + z)(x + y + z')(x+y + z)(x+y’+ z)(x + y' + z)(x'+y'+z) = (x + y + z)(x + y + z') (x + y' + z) (x'+y'+z) = M0 M1 M2 M6
= Π (0,1,2,6)
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STANDARD FORMS There are two types of standard forms:
the sum of products (SOP) The product of sums (POS).
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Sum of Products The sum of products is a Boolean
expression containing AND terms, called product terms, of one or more literals each. The sum denotes the ORing of these terms.
F = xy + z +xy'z'. (SOP)
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Product of Sums A product of sums is a Boolean expression
containing OR terms, called sum terms. Each term may have any number of literals. The product denotes the ANDing of these terms.
F = z(x+y)(x+y+z) (POS) F = x (xy' + zy) (nonstandard form)
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