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Chapter 3 Boolean Algebra and Digital Logic
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Chapter 3 Objectives • Understand the relationship between Boolean logic and digital computer circuits. • Learn how to design simple logic circuits. • Understand how digital circuits work together to form complex computer systems. 2
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3.1 Introduction • In the latter part of the nineteenth century, George Boole incensed philosophers and mathematicians alike when he suggested that logical thought could be represented through mathematical equations. – How dare anyone suggest that human thought could be encapsulated and manipulated like an algebraic formula? • Computers, as we know them today, are implementations of Boole’s Laws of Thought. – John Atanasoff and Claude Shannon were among the first to see this connection. 3
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George Boole, 1815 - 1864 • Born to working class parents • Taught himself mathematics and joined the faculty of Queen’s College in Ireland. • Wrote An Investigation of the Laws of Thought (1854) • Copyright © 2007 Elsevier Introduced binary variables 1-<4>
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3.2 Boolean Algebra • A Boolean operator can be completely described using a truth table. • The truth table for the Boolean operators AND and OR are shown at the right. • The AND operator is also known as a Boolean product. The OR operator is the Boolean sum. 5
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3.2 Boolean Algebra • The truth table for the Boolean NOT operator is shown at the right. • The NOT operation is most often designated by an overbar. It is sometimes indicated by a prime mark ( ‘ ) or an “elbow” (). 6
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3.2 Boolean Algebra • A Boolean function has: • • • At least one Boolean variable, At least one Boolean operator, and At least one input from the set {0,1}. • It produces an output that is also a member of the set {0,1}. Now you know why the binary numbering system is so handy in digital systems. 7
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Slide #8.

3.2 Boolean Algebra • The truth table for the Boolean function: is shown at the right. • To make evaluation of the Boolean function easier, the truth table contains extra (shaded) columns to hold evaluations of subparts of the function. 8
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3.2 Boolean Algebra • As with common arithmetic, Boolean operations have rules of precedence. • The NOT operator has highest priority, followed by AND and then OR. • This is how we chose the (shaded) function subparts in our table. 9
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3.2 Boolean Algebra • Most Boolean identities have an AND (product) form as well as an OR (sum) form. We give our identities using both forms. Our first group is rather intuitive: 10
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3.2 Boolean Algebra • Our second group of Boolean identities should be familiar to you from your study of algebra: 11
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3.2 Boolean Algebra • Our last group of Boolean identities are perhaps the most useful. • If you have studied set theory or formal logic, these laws are also familiar to you. 12
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3.2 Boolean Algebra • We can use Boolean identities to simplify: as follows: 13
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3.2 Boolean Algebra • Sometimes it is more economical to build a circuit using the complement of a function (and complementing its result) than it is to implement the function directly. • DeMorgan’s law provides an easy way of finding the complement of a Boolean function. • Recall DeMorgan’s law states: 14
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3.2 Boolean Algebra • DeMorgan’s law can be extended to any number of variables. • Replace each variable by its complement and change all ANDs to ORs and all ORs to ANDs. • Thus, we find the the complement of: is: 15
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3.2 Boolean Algebra • Through our exercises in simplifying Boolean expressions, we see that there are numerous ways of stating the same Boolean expression. – These “synonymous” forms are logically equivalent. – Logically equivalent expressions have identical truth tables. • In order to eliminate as much confusion as possible, designers express Boolean functions in standardized or canonical form. 16
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Slide #17.

3.2 Boolean Algebra • There are two canonical forms for Boolean expressions: sum-of-products and product-of-sums. – Recall the Boolean product is the AND operation and the Boolean sum is the OR operation. • In the sum-of-products form, ANDed variables are ORed together. – For example: • In the product-of-sums form, ORed variables are ANDed together: – For example: 17
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Slide #18.

3.2 Boolean Algebra • It is easy to convert a function to sum-of-products form using its truth table. • We are interested in the values of the variables that make the function true (=1). • Using the truth table, we list the values of the variables that result in a true function value. • Each group of variables is then ORed together. 18
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3.2 Boolean Algebra • The sum-of-products form for our function is: We note that this function is not in simplest terms. Our aim is only to rewrite our function in canonical sum-of-products form. 19
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3.3 Logic Gates • We have looked at Boolean functions in abstract terms. • In this section, we see that Boolean functions are implemented in digital computer circuits called gates. • A gate is an electronic device that produces a result based on two or more input values. – In reality, gates consist of one to six transistors, but digital designers think of them as a single unit. – Integrated circuits contain collections of gates suited to a particular purpose. 20
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Slide #21.

3.3 Logic Gates • The three simplest gates are the AND, OR, and NOT gates. • They correspond directly to their respective Boolean operations, as you can see by their truth tables. 21
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3.3 Logic Gates • Another very useful gate is the exclusive OR (XOR) gate. • The output of the XOR operation is true only when the values of the inputs differ. Note the special symbol  for the XOR operation. 22
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3.3 Logic Gates • NAND and NOR are two very important gates. Their symbols and truth tables are shown at the right. 23
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3.3 Logic Gates • NAND and NOR are known as universal gates because they are inexpensive to manufacture and any Boolean function can be constructed using only NAND or only NOR gates. 24
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3.3 Logic Gates • Gates can have multiple inputs and more than one output. – A second output can be provided for the complement of the operation. – We’ll see more of this later. 25
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3.3 Logic Gates • The main thing to remember is that combinations of gates implement Boolean functions. • The circuit below implements the Boolean function: We simplify our Boolean expressions so that we can create simpler circuits. 26
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source gate drain Transistor Function Polysilicon SiO2 n n p substrate g=0 gate source drain nMOS gate drain ON s s s s s s g Polysilicon d OFF g pMOS source d d nMOS g=1 OFF ON d d d SiO2 p p n substrate gate source 1-<27> drain Copyright © 2007 Elsevier
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Transistor Function • nMOS transistors pass good 0’s, so connect source to GND • pMOS transistors pass good 1’s, so connect source to VDD pMOS pull-up network inputs output nMOS pull-down network 1-<28> Copyright © 2007 Elsevier
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Slide #29.

CMOS Gates: NOT Gate NOT A VDD Y A Y=A A 0 1 Y 1 0 P1 Y N1 GND A P1 N1 Y 0 1 Copyright © 2007 Elsevier 1-<29>
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CMOS Gates: NOT Gate NOT A VDD Y A Y=A A 0 1 Y 1 0 Copyright © 2007 Elsevier P1 Y N1 GND A P1 N1 Y 0 ON OFF 1 1 OFF ON 0 1-<30>
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Slide #31.

CMOS Gates: NAND Gate Vdd NAND A B P2 Y Y = AB A 0 0 1 1 B 0 1 0 1 Y 1 1 1 0 A B P1 P1 A N1 B N2 P2 N1 Y N2 Y 0 0 0 1 1 0 1 1 Copyright © 2007 Elsevier 1-<31>
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Slide #32.

CMOS Gates: NAND Gate Vdd NAND A B P2 Y Y = AB A 0 0 1 1 B 0 1 0 1 Y 1 1 1 0 P1 A N1 B N2 Y A B P1 P2 N1 0 0 ON ON OFF OFF 1 0 1 ON OFF OFF ON 1 0 OFF ON ON 1 1 OFF OFF ON Copyright © 2007 Elsevier N2 Y 1 OFF 1 ON 0 1-<32>
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Slide #33.

NOR Gate How do you build a three-input NOR gate? 1-<33> Copyright © 2007 Elsevier
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NOR3 Gate Three-input NOR gate Vdd A B C Y 1-<34> Copyright © 2007 Elsevier
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Other CMOS Gates How do you build a two-input AND gate? 1-<35> Copyright © 2007 Elsevier
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Slide #36.

Other CMOS Gates Two-input AND gate A B Y 1-<36> Copyright © 2007 Elsevier
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Slide #37.

Logic Family Examples Logic VDD Family TTL VIL VI VO VO 5 (4.75 0. - 5.25) 8 H L H 2. 0. 2. 0 4 4 CMOS 5 (4.5 - 1. 3. 0. 3. 6) 35 15 33 84 LVTT L 3.3 (3 - 0. 3.6) 8 2. 0. 2. 0 4 4 LVCM 3.3 (3 - 0. OS 3.6) 9 1. 0. 2. 8 36 7 Copyright © 2007 Elsevier 1-<37>
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Slide #38.

RTL Logic 1-<38>
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