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ECE 15A
Fundamentals of Logic Design
Lecture 2
Malgorzata MarekSadowska
Electrical and Computer Engineering Department
UCSB
2
Today: The Algebra of Sets
±
The Algebra of Sets is
²
an example of Boolean Algebra
²
useful in manipulating digital circuits
±
We will investigate the nature of sets and the
way in which they may be combined
3
Definitions
±
Elements are basic objects
±
Collections of objects constitute sets
Basic objects
Example:
S
1
S
2
S
3
S
4
Sets
S
5
Universal set, denoted by “1”:
Consists of all elements
under consideration
Null set, denoted by “0”,
Contains no elements
4
Sets
±
Specification of
a set: enumeration
²
Example: {1,3,5,7}
±
Order of set elements is arbitrary
²
S1 = {pink, blue, red}
²
S2 = {red, blue, pink}
²
S1 = S2
±
A set is an unordered collection of elements
±
The algebra will be developed as an algebra for
sets, not for elements of sets
5
Empty set, Universal set
±
A set which contains no elements is called the
empty set, or the Null set.
²
Often denoted as 0, { }.
²
We will denote it as 0.
²
For any element x
0. [x is not an element of 0]
±
A set which is a collection of all elements
under consideration forms the universal set
²
We will denote this set as 1.
∈
6
Subsets
±
If sets A and B have the same elements: A=B;
otherwise A = B.
±
If all elements of A are elements of B, A is a
subset of B: A
B. B is a superset of A.
±
If all elements of A are elements of B and
there is an element in B which is not in A:
A is a proper subset of B: A
B.
⊆
⊂
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7
Examples
±
Every set is a subset of 1; 0 is a subset of every other set.
±
0 and 1 are not numbers!
Basic objects
Example, continues:
S
1
S
2
S
3
S
4
S
5
Universal set 1
m
m is a member of S
4
S
2
S
1
⊂
m
S
4
∈
⊆
S
2
S
1
S
4
S
5
=
p
={p}
S
3
r
k
u
= {r,k,u}
S
2
S
6
S’
6
(also denoted as S
6
)
Complement
of a set
S
6
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Example: red, black and yellow books
The red (R) books and some of the
black books (B) are in English (E);
The reminder of the black books (B) are in
German (G); Yellow books (Y) are in French (F).
Y
R
B
E
G
F
Y=F
⊇
B
G
E
R
⊇
9
Rules of forming new sets
±
The union
of sets X and Y is a set X+Y consisting of
all elements which are either in X or in Y.
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This note was uploaded on 01/20/2011 for the course ECE 15A taught by Professor M during the Winter '08 term at UCSB.
 Winter '08
 M
 Gate

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