Discrete Systems I
Lecture 01
The Basic Functions of Boolean Algebra
Profs. Koike and Yukita
Preparations for Lesson 01
Collection of Key Words
• Boolean Algebra ブール代数
• 1 (TRUE) と 0 (FALSE) の algebra
• Boolean functions ブール代数における演算
– And かつ
– Or または
– Not 否定
2
Preparation Task 1
Truth Table とは何か調べてみよ。
1, 0 の代わりに T, Fと表記することも多い
。
AND
NOT
x
y
OR
x y
x
y
x y
0 0
0
0 0
0
x
x
0 1
0
0 1
1
0 1
1 0
0
1 0
1
1 0
1 1
1
1 1
1
3
Preparation Task 2
Fill in the blanks below.
x y
x
y
x y
x y
x
y
x y
x y
x
y
x y
x y
x
y
0 0
0 0
0 0
0 0
0 1
0 1
0 1
0 1
1 0
1 0
1 0
1 0
1 1
1 1
1 1
1 1
x y
4
Symbolic Logic 形式論理
Let x and y denote propositio ns. The value of a propositio n (命題)
is either True or False. From now on we denote True and False
by 1 and 0, respective ly. The algebra of propositio ns is called
symbolic logic. We will study Boolean algebra , one of the basic
systems of symbolic logic.
Any algebra has its set of elements (要素、元 ) and operations (演算).
Think of arithmetic or high school algebra in which w e have operations
such as Add, Subtract, Multiply, and Divide on numbers, polynomial s,
and other symbolic expression s.
Boolean algebra has three basic operations : And, Or, and Not.
These are called Boolean functions.
5
The AND function
The AND function takes two arguments x and y, and
returns 0 or 1 according to the truth table given below.
x
y
x y
0 0
0
0 1
0
1 0
0
1 1
1
6
The AND gate electronically emulates
the AND function.
input
output
x
x y
y
7
The OR function
The AND function takes two arguments x and y, and
returns 0 or 1 according to the truth table given below.
x
y
x y
0 0
0
0 1
1
1 0
1
1 1
1
8
The OR gate electronically emulates
the OR function.
input
output
x
x y
y
9
The NOT function
The NOT function takes one argument x, and inverts the
value of x.
x
x
0 1
1 0
10
The NOT gate electronically emulates
the NOT function.
input
output
x
x
11
Bubble Representation of NOT gate
=
=
Exercise 1.1
A. What is the value of the output bit in each of the
following circuit diagrams?
13
Exercise 1.1
A. What is the value of the output bit in each of the
following circuit diagrams?
14
Exercise 1.1
B. If possible, determine the value of the missing input bit
in each of the following circuit diagrams.
15