The document summarizes key points from a lecture on Boolean algebra: - It discusses theorems and rules in Boolean algebra, including DeMorgan's Theorems, and the universality of NAND and NOR gates. - Specific topics covered include the commutative, associative, and distributive laws, truth tables, and examples applying Boolean algebra rules and identities to simplify expressions. - DeMorgan's Theorems allow the exchange of ANDs and ORs by inverting variables, and are useful in digital circuit design.

1. Points Addressed in this Lecture
• Theorems & rules in Boolean algebra
• DeMorgan’s Theorems
• Universality of NAND & NOR gates
Lecture 4: Boolean Algebra
• Active low & Active high
Professor Peter Cheung • Digital Integrated Circuits
Department of EEE, Imperial College London
(Floyd 4.1-4.4, 5.2-5.4)
(Tocci 3.8-3.14)
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Laws of Boolean Algebra
Commutative Laws of Boolean Algebra
• Commutative Laws A+B=B+A
• Associative Laws
• Distributive Law
A•B=B•A
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2. Associative Laws of Boolean Algebra Distributive Laws of Boolean Algebra
A • (B + C) = A • B + A • C
A + (B + C) = (A + B) + C
A (B + C) = A B + A C
A • (B • C) = (A • B) • C
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Rules of Boolean Algebra Rules of Boolean Algebra
• Rule 1
OR Truth Table
• Rule 2
• Rule 3
AND Truth Table
• Rule 4
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3. Rules of Boolean Algebra Rules of Boolean Algebra
• Rule 5 • Rule 9
• Rule 10: A + AB = A
OR Truth Table
• Rule 6
AND Truth Table OR Truth Table
• Rule 7
AND Truth Table
• Rule 8
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Rules of Boolean Algebra Examples
• Rule 11: A + AB = A + B
• Simplify the expression y = ABD + ABD
y = AB
• Simplify ( )
z = A + B (A + B )
z=B
• Rule 12: (A + B)(A + C) = A + BC
• Simplify x = ACD+ ABCD
x = ACD + BCD
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4. DeMorgan’s Theorems
Review Questions
• Theorem 1
y = AC + AB C
• Simplify (x + y ) = x ⋅ y Remember:
y = AC “Break the bar,
• Theorem 2
change the operator”
• Simplify y = ABC D + ABC D (x ⋅ y ) = x + y
y = AB D
– DeMorgan's theorem is very useful in digital circuit design
– It allows ANDs to be exchanged with ORs by using invertors
• Simplify y = AD + ABD – DeMorgan's Theorem can be extended to any number of
variables.
y = AD + BD
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Example of DeMorgan’s Theorem Example
• Simplify the expression
F = X .Y + P.Q 2 NAND plus 1 OR
1 OR with some input invertors
( )(
z = A+ C ⋅ B + D )
= X +Y + P +Q
to one having only single variables inverted.
z = AC + B D
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5. Implications of DeMorgan’s Theorems(I) Implications of DeMorgan’s Theorems(II)
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Example Review Questions
• Determine the output expression for the below • Using DeMorgan’s Theorems to convert the
circuit and simplify it using DeMorgan’s expressions to one that has only single-variable
Theorem inversions.
y = R ST + Q y = ( R + S + T )Q
z = ( A + B )⋅ C z = AB + C
• Use DeMorgan’s theorems to convert below
expression to an expression containing only
single-variable inversions.
y = A + B + CD y = A B(C + D )
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6. Universality of NAND gates Universality of NOR gate
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Basic Characteristics of Digital ICs
• Digital ICs (chips): a collection of resistors, diodes and transistors
fabricated on a single piece of semiconductor materials called
substrate.
• Dual-in-line package (DIP) is a common type of packages.
Logic probe is used to monitor the logic level activity at an IC
pin or any other accessible point in a logic circuit
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7. Example
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Alternate Logic-Gate Representations Obtain alternative symbol from standard ones
• Invert each input and output of the standard
symbol. This is done by adding bubbles(small
circles) on input and output lines that do not
have bubbles and by removing bubbles that are
already there.
• Change the operation symbol from AND to OR,
or from OR to AND. (In the special case of the
INVERTER, the operation symbol is not
changed.)
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8. Several points Logic-symbol interpretation
• The equivalences can be extended to gates with any
number of inputs. • Active high/low
• None of the standard symbols have bubbles on their – When an input or output line on a logic circuit
inputs, and all the alternate symbols do. symbol has no bubble on it, that line is said to
• The standard and alternate symbols for each gate be active-high, otherwise it is active-low.
represent the same physical circuit; there is no
difference in the circuits represented by the two
symbols.
• NAND and NOR gates are inverting gates, and so
both the standard and the alternate symbols for each
will have a bubble on either the input or the output,
AND and OR gates are non-inverting gates, and so
the alternate symbols for each will have bubbles on
both inputs and output.
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Interpretation of the two NAND gate symbols Interpretation of the two OR gate symbols
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9. Equivalency of three circuits
• Which circuit diagram should be used?
• Bubble placement
Whenever possible, choose gate symbols so
that bubble outputs are connected to bubble
inputs, and nonbubble outputs to nonbubble
inputs
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Active High & Active Low Representations
• When a logic signal is in its active state, it can
be said to be asserted (=activated). When a
logic signal is not in its active state, it is said
to be unasserted (=inactive).
• The overbar is simply a way to emphasize
that these are active-LOW signals.
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10. Summary: Summary
How to represent the basic logic functions • Boolean Algebra: a mathematical tool used in the analysis
and design of digital circuits
• Logical statements in our own language • OR, AND, NOT: basic Boolean operations
• Truth tables • OR: HIGH output when any input is HIGH
• AND: HIGH output only when all inputs are HIGH
• Traditional graphic logic symbols • NOT: output is the opposite logic level as the input
• Boolean algebra expressions • NOR: OR with its output connected to an INVERTER
• Timing diagrams • NAND: AND with its output connected to an INVERTER
• Boolean theorems and rules: to simplify the expression of a
logic circuit and can lead to a simpler way of implementing
the circuit
• NAND, NOR: can be used to implement any of the basic
Boolean operations
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