Discrete Systems I
Lecture 02
Combinational Logic
Profs. Koike and Yukita
Preparations for Lesson 02
Collection of Key Expressions
• Any function is realized in terms of primitive
functions.
• There are several different ways of connecting
primitive functions to create the same
function.
• These are called different realizations of a
function.
2
Preparation Task 1
Complete the truth tab le for each of the following expression s.
x y z ( x  y)  z
0 0 0
x y z ( x  y)  ( x  z)
0 0 0
0 0 1
0 0 1
0 1 0
0 1 0
0 1 1
0 1 1
1 0 0
1 0 0
1 0 1
1 0 1
1 1 0
1 1 0
1 1 1
1 1 1
3
Preparation Task 2
Find a circuit diagram that corresponds to each
function given in Task 1.
4
AND and NOT
x
y
x y
Read as “NOT x AND y”
5
Step by step construction of a truth
table
x
y
x
0 0 1
x
y
x
x y
x
y
x y
0 0 1
0
0 0
0
0 1 1  0 1 1
1
 0 1
1
1
0
1
0
0
0
1 1
0
0 0
1 1
0
1
0 0
1 1
0
6
Grouping with parentheses
( x  y)  x
7
Step by step construction of a truth
table
x
y
x y
x  y ( x  y)  x
0 0
0
1
1
0 1
0
1
1
1
0
0
1
1
1 1
1
0
1
8
With split y
( x  y)  ( y  z)
9
Step by step construction of a truth
table
x y
y
0 0 0
0
1
1
0
0 0 1
0
1
1
0
0 1 0
1
0
0
0
0 1 1
1
0
1
1
1 0 0
1
1
1
1
1 0 1
1
1
1
1
1 1 0
1
0
0
0
1 1 1
1
0
1
1
x
y
z
y  z ( x  y)  ( y  z)
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Exercise 2.1A
For each of the following Boolean algebraic expressions:
(a) draw the corresponding circuit diagram and
(b) write the truth table. Use extra columns in the truth tables
to keep track of intermediate results.
x y
6. ( x  y )  ( y  z )
2. ( x  y )  z
7. ( x  y )  ( x  y )
3. ( x  y )  x
8. x  y
4. x  x
9. (( a  b)  c)  (c  d )
1.
5. x  ( y  ( x  z ))
10.
(( x  y )  ( x  z ))  ( x  ( y  z ))
11.
( x  y )  (( x  y )  ( x  y ))
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Exercise 2.1B
For each of the following circuit diagrams:
(a) write the corresponding Boolean algebraic expression and
(b) write the truth table. Use extra columns in the truth tables
to keep track of intermediate results.
12.
14.
13.
15.
12
Exercise 2.1B continued
16.
17.
18.
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Exercise 2.1D (C is intentionally skipped)
Write the algebraic expression s of the functions whose truth tab les are
presented in 20, 21, and 22.
20.
x
21.
y
x
y
22.
x
y
0 0 0
0 0 0
0 0 1
0 1 1
0 1 0
0 1 0
1 0 0
1 0 1
1 0 0
1 1 0
1 1 1
1 1 1
Notation: Generalized AND and OR
that take more than two inputs.
Is defined by
Is defined by
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Example
( x  y  z)  ( x  y  z)  ( x  y)
16
Truth table for the example
x y
x y z
x y z
0 0 0 1 1
0
0
1
1
0
0 0 1 1 0
0
1
0
1
0
0 1 0 0 1
0
1
1
1
1
0 1 1 0 0
0
1
1
1
1
1 0 0 1 1
1
1
1
0
0
1 0 1 1 0
1
1
1
0
0
1 1 0 0 1
0
1
1
1
1
1 1 1 0 0
0
1
1
1
1
x
y
z
y
z
x  y ( x  y  z)  ( x  y  z)  ( x  y)
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Exercise 2.2
A. For each of the Boolean algebraic expression s : (a) draw the
correspond ing circuit diagram and (b) write the truth tab le.
Use extra columns in the truth tab les to keep track of
intermedia te results.
1. a  b  c  d
4.
( x  y  z)  ( x  z)
2. (a  b  c)  (b  c)  (a  c)
5.
(a  b  c)
3. ( x  y )  ( x  y )  ( x  y )
6.
(a  b  c)  a  b
B. Compare the outputs of the functions given in 5 and 6.
Interpret the result.
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Exercise 2.2
C. For each of the following circuit diagram write (a) the
correspond ing Boolean algebraic expression s and (b) the truth tab le.
Use extra columns in the truth tab les to keep track of intermedia te results.
D. Compare the outputs of the functions given in 7, 8, and 9.
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