Download (why?) 1.6. Mathematical Structures

Boolean Matrix
Zero-one Matrices
 All entries are 0 or 1.
 Operations are  and .
 Boolean product is defined using:
 for multiplication, and
 for addition.
Boolean Operations
1 0 1
0 1 0 
A
B


0
1
0
1
1
0




0
A  B called "meet" A  B  
0
1
A  B called " join" A  B  
1
0
1
1
1
0
0
1
0
Boolean Product
A  aij  be m  k , and B  bij  be k  n
A
B=
c  ,
ij
cij  (ai1  b1 j )  (ai 2  b2 j )    (aik  bkj )
1 0
1 1 0


A 0 1 , B



0
1
1


1 0
A
B=
Since this is “or’d”, you
can stop when you find
a ‘1’
(1  1)  (0  0) 1 0 1 1 0
 (0  1)  (1  0) 1 1  0 1 1
(1  1)  (0  0) 1 0 1 1 0
Boolean Product Properties
• In
general, A
• Example
BB
A
1 1 0
0 0 1
A  0 1 1 , B  0 0 1




1 1 0
1 1 1
A
B =
0 0 1
1 1 1


0 0 1
B
A
1 1 0
 1 1 0
1 1 1
1.6. Mathematical Structures
 Mathematical structure (system)
Such a collection of objects with operations defined on them and
the accompanying properties form a mathematical structure or
system, for instance,
Example 1: The collection of sets with the operations of union,
intersection and complement and their accompanying properties
is a mathematical structure. Denoted by
(sets, U, ∩ , -)
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1.6. Mathematical Structures
 Binary operation
An operation that combines two objects
 Unary operation
An operation that requires only one object
Example: the structure (5x5 matrices, +, *, T)
the operations + and * are binary operations
the operation T is a unary operation
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1.6. Mathematical Structures
 Closure
A structure is closed with respect to an operation if that
operation always produces another/same member of the
collection of objects.
Example 3: The structure (5x5 matrices, +, *, T) is closed with
respect to +, * and T. (why?)
Example 4: The structure (odd integers, +, *) is closed with
respected to *, while it is not closed with respected to +. (why?)
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1.6. Mathematical Structures
 Commutative property
If the order of the objects does not affect the outcome of a binary
operation, we say that the operation is commutative , namely
if x □ y = y □ x, where □ is some binary operation with
commutative property.
Example 6
(a) Join and meet for Boolean matrices are commutative operations
A V B =B V A and A ^ B = B ^ A
(b) Ordinary matrix multiplication is not a commutative operation.
AB ≠ BA
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1.6. Mathematical Structures
 Associative property
if □ is a binary operation, then □ is associative or has
associative property if
(x □ y) □ z = x □ (y □ z)
Example 7
Set union is an associative operation, since
(A U B) U C = A U (B U C) is always true
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1.6. Mathematical Structures
 Distributive property
If a mathematical structure has tow binary operations, say □ and
∇, a distributive property has the following pattern:
x □ (y ∇ z) = (x □ y) ∇ ( x □ z )
we say that □ distributes over ∇
Example 8 (b)
the structure (sets, U, ∩, -) has two distributive properties:
A U (B ∩ C) =(A U B) ∩ (A U C)
A ∩ (B U C) =(A ∩ B) U (A ∩ C)
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1.6. Mathematical Structures
 De Morgan’s law
If the unary operation is ○ and the binary operation □ and ∇,
then De Morgan’s law are
(x □ y) ○ =x ○ ∇ y ○ , (x ∇ y) = x ○ □ y ○
Example 9
(a) Union, intersection and complement
(b) The structure (real numbers, +, *, sqrt) does not satisfy De
Morgan’s law (why?)
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1.6. Mathematical Structures
 Identify
If a structure with a binary operation □ contain an element e,
satisfying that
x□e=e□x=x
for all x in the collection
we call e an identify for the operation □
Example 10:
For (n-by-n matrices, +,*, T), In is the identify for matrix
multiplication and the n-by-n zero matrix is the identify matrix
addition.
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1.6. Mathematical Structures
 Theorem 1: If e is an identify for a binary
operation □, then e is unique.
Proof:
Assume i is another object with identify property, then we have i
□ e = e □ i = e; since e is also an identify for □, then we have i
□ e = e □ i = i, therefore e = i, which means that there is at most
one object with the identify property for □.
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1.6. Mathematical Structures
 Inverse
If a binary operation □ has an identity e, we say y is a □inverse of x if x □y=y □x=e
Example 11:
(a) In the structure (3-by-3 matrices, +, *, T), each matrix A=[aij]
has +-inverse(additive inverse), -A=[-aij]. (why ?)
(b) In the structure (integers, +, *), only the integers 1 and -1
have multiplicative inverses. (why?)
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 Theorem 2: If □ is an associative operation and
x has a □-inverse y, then y is unique.
Proof:
Assume there is another □-inverse for x, say z, then
(z □ x) □ y = e □ y = y, and z □ (x □ y) =z □ e =z
since □ us associative, (z □ x) □ y = z □ (x □ y) and so
y=z, which means that y is unique.
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