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Chapter 3: Boolean Algebra and Digital Logic

• Chapter 3: Digital Logic – 1

CS140 Computer Organization

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.

• Chapter 3: Digital Logic – 2

3.2 Boolean Algebra

• I’m assuming that you have taken or are currently taking Discrete Math. So I’m not planning on talking about Boolean algebra other than to connect it with circuits.
• The Slides written by Null & Lobur have been moved to an Appendix at the end of this set. - Chapter 3: Digital Logic – 4

3.3 Logic Gates

• 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.

• Chapter 3: Digital Logic – 5

Vs – is ground = 0 Volts

Vd – high voltage – for all the

things we’re doing, this is

+5V, but there are many

possibilities.

Vg – gate voltage – depending

on this value, the electrons

can or can not flow from

high to low voltage.

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.

• And these representations map exactly into the transistors

on the last two slides.

• Chapter 3: Digital Logic – 7 74LS Quad 2-input AND

3.3 Logic Gates

• The output of the XOR operation

is true only when the values of

the inputs differ.

• Chapter 3: Digital Logic – 8

Note the special symbol 

for the XOR operation.

• Symbols for NAND and NOR, and

truth tables are shown at the right.

74LS Quad 2-input NOR

3.4 Digital Components

• Combinations of gates implement Boolean functions.
• The circuit below implements the function:
  • Chapter 3: Digital Logic – 10
• This is an example of a combinational logic circuit.
• Combinational logic circuits produce a specified output

(almost) at the instant when input values are applied.

• Later we’ll explore circuits where this is not the case.

3.5 Combinational Circuits

• As we see, the sum can be found using the XOR operation and the carry using the AND operation. - Chapter 3: Digital Logic – 11
• Combinational logic circuits give us many useful devices.
• One of the simplest is the half adder , which finds the sum of two bits.
• We can gain some insight as to the construction of a half adder by looking at its truth table, shown at the right.

3.5 Combinational Circuits

• Just as we combined half adders to make a full adder,

full adders can connected in series.

• The carry bit “ripples” from one adder to the next;

hence, this configuration is called a ripple-carry adder.

• Chapter 3: Digital Logic – 13 74LS

This is a 4-bit adder

that you can program

as part of your Project.

3.5 Combinational Circuits

• Decoders are another important type of combinational circuit.
• Among other things, they are useful in selecting a memory location based on

a binary value placed on the address lines of a memory bus.

• Address decoders with n inputs can select any of 2 n

locations.

• Chapter 3: Digital Logic – 14
• This is what a 2-to-4 decoder looks like on the inside.

If x = 0 and y = 1,

which output line

is enabled?

3.5 Combinational Circuits

• Chapter 3: Digital Logic – 16 74LS One of Ten Decoder

3.5 Combinational Circuits

• A multiplexer does just the opposite of a decoder.
• It selects a single output from several inputs.
• The particular input chosen for output is determined by the value of the multiplexer’s control lines.
• To be able to select among n inputs, log

n control lines are needed.

• Chapter 3: Digital Logic – 17

This is a block

diagram for a

multiplexer.

3.5 Combinational Circuits

• Chapter 3: Digital Logic – 19

3.5 Combinational Circuits

• This shifter

moves the bits

of a nibble one

position to the

left or right.

• Chapter 3: Digital Logic – 20

If S = 0, in which

direction do the

input bits shift?