CS1104-3 Lecture 3: Boolean Algebra 2
Lecture 3: Boolean Algebra
Digital circuits
Boolean Algebra
Two-Valued Boolean Algebra
Boolean Algebra Postulates
Precedence of Operators
Truth Table & Proofs
Duality
CS1104-3 Lecture 3: Boolean Algebra 3
Lecture 3: Boolean Algebra
Basic Theorems of Boolean Algebra
Boolean Functions
Complement of Functions
Standard Forms
Minterm & Maxterm
Canonical Forms
Conversion of Canonical Forms
Binary Functions
CS1104-3 Introduction 4
IntroductionBoolean algebra forms the basis of logic circuit design. Consider very simple but common example: if (A is true) and (B is false) then print “the solution is found”. In this case, two Boolean expressions (A is true) and (B is false) are related by a connective ‘and’. How do we define these? This and related things are discussed in this chapter.
In typical circuit design, there are many conditions to be taken care of (for example, when the ‘second counter’ = 60, the ‘minute counter’ is incremented and ‘second counter’ is made 0. Thus it is quite important to understand Boolean algebra. In subsequent chapters, we are going to further study how to minimize the circuit using laws of Boolean algebra (that is very interesting…)
CS1104-3 Digital Circuits 5
Digital Circuits (1/2)
Digital circuit can be represented by a black-box with inputs on one side, and outputs on the other.
The input/output signals are discrete/digital in nature, typically with two distinct voltages (a high voltage and a low voltage).
In contrast, analog circuits use continuous signals.
Digital circuit
inputs outputs: :
High
Low
CS1104-3 Digital Circuits 6
Digital Circuits (2/2)
Advantages of Digital Circuits over Analog Circuits: more reliable (simpler circuits, less noise-prone) specified accuracy (determinable) but slower response time (sampling rate)
Important advantages for two-valued Digital Circuit:
Mathematical Model – Boolean Algebra
Can help design, analyse, simplify Digital Circuits.
CS1104-3 Boolean Algebra 7
Boolean Algebra (1/2)
Boolean Algebra named after George Boole who used it to study human logical reasoning – calculus of proposition.
Events : true or false
Connectives : a OR b; a AND b, NOT a
Example: Either “it has rained” OR “someone splashed water”, “must be tall” AND “good vision”.
What is an Algebra? (e.g. algebra of integers)set of elements (e.g. 0,1,2,..)set of operations (e.g. +, -, *,..)postulates/axioms (e.g. 0 + x = x,..)
CS1104-3 Boolean Algebra 8
Boolean Algebra (2/2)
a b a AND bF F FF T FT F FT T T
a b a OR bF F FF T TT F TT T T
a NOT aF TT F
Later, Shannon introduced switching algebra (two-valued Boolean algebra) to represent bi-stable switching circuit.
CS1104-3 Two-valued Boolean Algebra 9
Two-valued Boolean Algebra Set of elements: {0,1}
Set of operations: { ., + , ¬ }
x y x . y0 0 00 1 01 0 01 1 1
x y x + y0 0 00 1 11 0 11 1 1
x ¬x0 11 0
Signals: High = 5V = 1; Low = 0V = 0
x
yx.y
x
yx+y x x'
Sometimes denoted by ’, for example a’
CS1104-3 Boolean Algebra Postulates 10
Boolean Algebra Postulates (1/3)
The set B contains at least two distinct elements x and y.
Closure: For every x, y in B, x + y is in B x . y is in B
Commutative laws: For every x, y in B, x + y = y + x
x . y = y . x
A Boolean algebra consists of a set of elements B, with two binary operations {+} and {.} and a unary operation {'}, such that the following axioms hold:
CS1104-3 Boolean Algebra Postulates 11
Boolean Algebra Postulates (2/3)
Associative laws: For every x, y, z in B, (x + y) + z = x + (y + z) = x + y + z (x . y) . z = x .( y . z) = x . y . z
Identities (0 and 1): 0 + x = x + 0 = x for every x in B 1 . x = x . 1 = x for every x in B
Distributive laws: For every x, y, z in B, x . (y + z) = (x . y) + (x . z) x + (y . z) = (x + y) . (x + z)
CS1104-3 Boolean Algebra Postulates 12
Boolean Algebra Postulates (3/3)
Complement: For every x in B, there exists an element x' in B such that x + x' = 1 x . x' = 0
The set B = {0, 1} and the logical operations OR, AND and NOT satisfy all the axioms of a Boolean algebra.
A Boolean function maps some inputs over {0,1} into {0,1}
A Boolean expression is an algebraic statement containing Boolean variables and operators.
CS1104-3 Precedence of Operators 13
Precedence of Operators (1/2)
To lessen the brackets used in writing Boolean expressions, operator precedence can be used.
Precedence (highest to lowest): ' . +
Examples:
a . b + c = (a . b) + c
b' + c = (b') + c
a + b' . c = a + ((b') . c)
CS1104-3 Precedence of Operators 14
Precedence of Operators (2/2)
Use brackets to overwrite precedence.
Examples:
a . (b + c)
(a + b)' . c
CS1104-3 Truth Table 15
Truth Table (1/2)
Provides a listing of every possible combination of inputs and its corresponding outputs.
Example (2 inputs, 2 outputs):
x y x . y x + y0 0 0 00 1 0 11 0 0 11 1 1 1
INPUTS OUTPUTS… …… …
CS1104-3 Truth Table 16
Truth Table (2/2)
Example (3 inputs, 2 outputs):
x y z y + z x.(y + z)0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 1 01 0 0 0 01 0 1 1 11 1 0 1 11 1 1 1 1
CS1104-3 Proof using Truth Table 17
Proof using Truth Table
Can use truth table to prove by perfect induction. Prove that: x . (y + z) = (x . y) + (x . z)(i) Construct truth table for LHS & RHS of above equality.
(ii) Check that LHS = RHSPostulate is SATISFIED because output column 2 & 5 (for
LHS & RHS expressions) are equal for all cases.
x y z y + z x.(y + z) x.y x.z (x.y)+(x.z)0 0 0 0 0 0 0 00 0 1 1 0 0 0 00 1 0 1 0 0 0 00 1 1 1 0 0 0 01 0 0 0 0 0 0 01 0 1 1 1 0 1 11 1 0 1 1 1 0 11 1 1 1 1 1 1 1
CS1104-3 Duality 19
Duality (1/2)
Duality Principle – every valid Boolean expression (equality) remains valid if the operators and identity elements are interchanged, as follows:
+ .1 0
Example: Given the expressiona + (b.c) = (a+b).(a+c)
then its dual expression isa . (b+c) = (a.b) + (a.c)
CS1104-3 Duality 20
Duality (2/2)
Duality gives free theorems – “two for the price of one”. You prove one theorem and the other comes for free!
If (x+y+z)' = x'.y.'z' is valid, then its dual is also valid:
(x.y.z)' = x'+y'+z’
If x + 1 = 1 is valid, then its dual is also valid:
x . 0 = 0
CS1104-3 Basic Theorems of Boolean Algebra
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Basic Theorems of Boolean Algebra (1/5)
Apart from the axioms/postulates, there are other useful theorems.
1. Idempotency.
(a) x + x = x (b) x . x = x
Proof of (a):
x + x = (x + x).1 (identity)
= (x + x).(x + x') (complementarity)
= x + x.x' (distributivity)
= x + 0 (complementarity)
= x (identity)
CS1104-3 Basic Theorems of Boolean Algebra
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Basic Theorems of Boolean Algebra (2/5)
2. Null elements for + and . operators.
(a) x + 1 = 1 (b) x . 0 = 0
3. Involution. (x')' = x
4. Absorption.
(a) x + x.y = x (b) x.(x + y) = x
5. Absorption (variant).
(a) x + x'.y = x+y (b) x.(x' + y) = x.y
CS1104-3 Basic Theorems of Boolean Algebra
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Basic Theorems of Boolean Algebra (3/5)
6. DeMorgan.
(a) (x + y)' = x'.y'
(b) (x.y)' = x' + y'
7. Consensus.
(a) x.y + x'.z + y.z = x.y + x'.z
(b) (x+y).(x'+z).(y+z) = (x+y).(x'+z)
CS1104-3 Basic Theorems of Boolean Algebra
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Basic Theorems of Boolean Algebra (4/5)
Theorems can be proved using the truth table method. (Exercise: Prove De-Morgan’s theorem using the truth table.)
They can also be proved by algebraic manipulation using axioms/postulates or other basic theorems.
CS1104-3 Basic Theorems of Boolean Algebra
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Basic Theorems of Boolean Algebra (5/5)
Theorem 4a (absorption) can be proved by:
x + x.y = x.1 + x.y (identity)
= x.(1 + y) (distributivity)
= x.(y + 1) (commutativity)
= x.1 (Theorem 2a)
= x (identity)
By duality, theorem 4b:
x.(x+y) = x
Try prove this by algebraic manipulation.
CS1104-3 Boolean Functions 26
Boolean Functions (1/2)
Boolean function is an expression formed with binary variables, the two binary operators, OR and AND, and the unary operator, NOT, parenthesis and the equal sign.
Its result is also a binary value.
We usually use . for AND, + for OR, and ' or ¬ for NOT. Sometimes, we may omit the . if there is no ambiguity.
CS1104-3 Boolean Functions 27
Boolean Functions (2/2)
Examples: F1= x.y.z' F2= x + y'.z F3=(x'.y'.z)+(x'.y.z)+(x.y') F4=x.y'+x'.z
x y z F1 F2 F3 F40 0 0 0 0 0 00 0 1 0 1 1 10 1 0 0 0 0 00 1 1 0 0 1 11 0 0 0 1 1 11 0 1 0 1 1 11 1 0 1 1 0 01 1 1 0 1 0 0
From the truth table, F3=F4.Can you also prove by algebraic manipulation that F3=F4?
CS1104-3 Complement of Functions 28
Complement of Functions (1/2)
Given a function, F, the complement of this function, F', is obtained by interchanging 1 with 0 in the function’s output values.
x y z F1 F1'0 0 0 0 10 0 1 0 10 1 0 0 10 1 1 0 11 0 0 0 11 0 1 0 11 1 0 1 01 1 1 0 1
Example: F1 = x.y.z'
Complement: F1' = (x.y.z')' = x' + y' + (z')' DeMorgan = x' + y' + z Involution
CS1104-3 Complement of Functions 29
Complement of Functions (2/2)
More general DeMorgan’s theorems useful for obtaining complement functions:
(A + B + C + ... + Z)' = A' . B' . C' . … . Z'
(A . B . C ... . Z)' = A' + B' + C' + … + Z'
CS1104-3 Standard Forms 30
Standard Forms (1/3)
Certain types of Boolean expressions lead to gating networks which are desirable from implementation viewpoint.
Two Standard Forms: Sum-of-Products and Product-of-Sums
Literals: a variable on its own or in its complemented form. Examples: x, x' , y, y'
Product Term: a single literal or a logical product (AND) of several literals.
Examples: x, x.y.z', A'.B, A.B, e.g'.w.v
CS1104-3 Standard Forms 31
Standard Forms (2/3)
Sum Term: a single literal or a logical sum (OR) of several literals.
Examples: x, x+y+z', A'+B, A+B, c+d+h'+j Sum-of-Products (SOP) Expression: a product term
or a logical sum (OR) of several product terms.
Examples: x, x+y.z', x.y'+x'.y.z, A.B+A'.B', A + B'.C + A.C' + C.D
Product-of-Sums (POS) Expression: a sum term or a logical product (AND) of several sum terms.
Examples: x, x.(y+z'), (x+y').(x'+y+z), (A+B).(A'+B'), (A+B+C).D'.(B'+D+E')
CS1104-3 Standard Forms 32
Standard Forms (3/3)
Every Boolean expression can either be expressed as sum-of-products or product-of-sums expression.
Examples:
SOP: x.y + x.y + x.y.z
POS: (x + y).(x + y).(x + z)both: x + y + z or x.y.zneither: x.(w + y.z) or z + w.x.y + v.(x.z + w)
CS1104-3 Minterm & Maxterm 33
Minterm & Maxterm (1/3)
Consider two binary variables x, y.
Each variable may appear as itself or in complemented form as literals (i.e. x, x' & y, y' )
For two variables, there are four possible combinations with the AND operator, namely:
x'.y', x'.y, x.y', x.y
These product terms are called the minterms.
A minterm of n variables is the product of n literals from the different variables.
CS1104-3 Minterm & Maxterm 34
Minterm & Maxterm (2/3)
In general, n variables can give 2n minterms.
In a similar fashion, a maxterm of n variables is the sum of n literals from the different variables.
Examples: x'+y', x'+y, x+y',x+y
In general, n variables can give 2n maxterms.
CS1104-3 Minterm & Maxterm 35
Minterm & Maxterm (3/3)
The minterms and maxterms of 2 variables are denoted by m0 to m3 and M0 to M3 respectively:
Minterms Maxterms x y term notation term notation 0 0 x'.y' m0 x+y M0 0 1 x'.y m1 x+y' M1 1 0 x.y' m2 x'+y M2 1 1 x.y m3 x'+y' M3
Each minterm is the complement of the corresponding maxterm: Example: m2 = x.y'
m2' = (x.y')' = x' + (y')' = x'+y = M2
CS1104-3 Canonical Form: Sum-of-Minterms
36
Canonical Form: Sum of Minterms (1/3)
What is a canonical/normal form? A unique form for representing something.
Minterms are product terms. Can express Boolean functions using Sum-of-Minterms
form.
CS1104-3 Canonical Form: Sum-of-Minterms
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Canonical Form: Sum of Minterms (2/3)
a) Obtain the truth table. Example:
x y z F1 F2 F30 0 0 0 0 00 0 1 0 1 10 1 0 0 0 00 1 1 0 0 11 0 0 0 1 11 0 1 0 1 11 1 0 1 1 01 1 1 0 1 0
CS1104-3 Canonical Form: Sum-of-Minterms
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Canonical Form: Sum of Minterms (3/3)
b) Obtain Sum-of-Minterms by gathering/summing the minterms of the function (where result is a 1)F1 = x.y.z' = m(6)
F2 = x'.y'.z + x.y'.z' + x.y'.z + x.y.z' + x.y.z = m(1,4,5,6,7)
F3 = x'.y'.z + x'.y.z + x.y'.z' +x.y'.z = m(1,3,4,5)
x y z F1 F2 F30 0 0 0 0 00 0 1 0 1 10 1 0 0 0 00 1 1 0 0 11 0 0 0 1 11 0 1 0 1 11 1 0 1 1 01 1 1 0 1 0
CS1104-3 Canonical Form: Product-of-Maxterms
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Canonical Form: Product of Maxterms (1/4)
Maxterms are sum terms.
For Boolean functions, the maxterms of a function are the terms for which the result is 0.
Boolean functions can be expressed as Products-of-Maxterms.
CS1104-3 Canonical Form: Product-of-Maxterms
40
Canonical Form: Product of Maxterms (2/4)
E.g.: F2 = M(0,2,3) = (x+y+z).(x+y'+z).(x+y'+z')
F3 = M(0,2,6,7)
= (x+y+z).(x+y'+z).(x'+y'+z).(x'+y'+z') x y z F1 F2 F30 0 0 0 0 00 0 1 0 1 10 1 0 0 0 00 1 1 0 0 11 0 0 0 1 11 0 1 0 1 11 1 0 1 1 01 1 1 0 1 0
CS1104-3 Canonical Form: Product-of-Maxterms
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Canonical Form: Product of Maxterms (3/4)
Why is this so? Take F2 as an example.
F2 = m(1,4,5,6,7)
The complement function of F2 is:
F2' = m(0,2,3)
= m0 + m2 + m3
(Complement functions’ minterms are the opposite of their original functions, i.e. when original function = 0)
x y z F2 F2'0 0 0 0 10 0 1 1 00 1 0 0 10 1 1 0 11 0 0 1 01 0 1 1 01 1 0 1 01 1 1 1 0
CS1104-3 Canonical Form: Product-of-Maxterms
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Canonical Form: Product of Maxterms (4/4)
From previous slide, F2' = m0 + m2 + m3
Therefore:
F2 = (m0 + m2 + m3 )'
= m0' . m2' . m3' DeMorgan
= M0 . M2 . M3 mx' = Mx
= M(0,2,3)
Every Boolean function can be expressed as either Sum-of-Minterms or Product-of-Maxterms.
CS1104-3 Quick Reviwe Questions (2) 43
Quick Review Questions (2)
Textbook pages 54-55.
Questions 3-2 to 3-11.
CS1104-3 Conversion of Canonical Forms 44
Conversion of Canonical Forms (1/3)
Sum-of-Minterms Product-of-Maxterms Rewrite minterm shorthand using maxterm shorthand. Replace minterm indices with indices not already used.
Eg: F1(A,B,C) = m(3,4,5,6,7) = M(0,1,2)
Product-of-Maxterms Sum-of-Minterms Rewrite maxterm shorthand using minterm shorthand. Replace maxterm indices with indices not already used.
Eg: F2(A,B,C) = M(0,3,5,6) = m(1,2,4,7)
CS1104-3 Conversion of Canonical Forms 45
Conversion of Canonical Forms (2/3)
Sum-of-Minterms of F Sum-of-Minterms of F' In minterm shorthand form, list the indices not already used
in F.
Eg: F1(A,B,C) = m(3,4,5,6,7)
F1'(A,B,C) = m(0,1,2)
Product-of-Maxterms of F Prod-of-Maxterms of F' In maxterm shorthand form, list the indices not already used
in F.
Eg: F1(A,B,C) = M(0,1,2)
F1'(A,B,C) = M(3,4,5,6,7)
CS1104-3 Conversion of Canonical Forms 46
Conversion of Canonical Forms (3/3)
Sum-of-Minterms of F Product-of-Maxterms of F' Rewrite in maxterm shorthand form, using the same indices
as in F.
Eg: F1(A,B,C) = m(3,4,5,6,7)
F1'(A,B,C) = M(3,4,5,6,7)
Product-of-Maxterms of F Sum-of-Minterms of F' Rewrite in minterm shorthand form, using the same indices
as in F.
Eg: F1(A,B,C) = M(0,1,2)
F1'(A,B,C) = m(0,1,2)
CS1104-3 Binary Functions 48
Binary Functions (1/2)
Given n variables, there are 2n possible minterms.
As each function can be expressed as sum-of-minterms, there could be 22n
different functions.
In the case of two variables, there are 22 =4 possible minterms; and 24=16 different possible binary functions.
The 16 possible binary functions are shown in the next slide.
CS1104-3 Binary Functions 49
Binary Functions (2/2)x y F0 F1 F2 F3 F4 F5 F6 F7
0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 1 1 1 11 0 0 0 1 1 0 0 1 11 1 0 1 0 1 0 1 0 1Symbol . / / +Name AND XOR OR
x y F8 F9 F10 F11 F12 F13 F14 F15
0 0 1 1 1 1 1 1 1 10 1 0 0 0 0 1 1 1 11 0 0 0 1 1 0 0 1 11 1 0 1 0 1 0 1 0 1Symbol ' ' Name NOR XNOR NAND