Some binary
operations on Boolean Matrices
A Boolean Matrix
is a real matrix whose entries are either 0 or 1.
Note
that the boolean entries 0 and 1 can be defined in several ways. In electrical
switch to describe “on and off”, in graph theory, the “adjacency matrix” etc ,
the boolean entries 0 and 1 are used. We consider the same type of Boolean
matrices in our discussion.
The
following two kinds of operations on the collection of all boolean matrices are
defined.
Let A = [aij]
and B = [bij] be any two boolean matrices of the same type.
Then their join
∨ and meet ∧ are defined as follows:
A ∨ B = [aij] ∨ [bij] = [aij ∨ bij] = [cij]
A ∧ B = [aij] ∧ [bij] = [aij
∧ bij] = [cij]
It is clear that ( a ∨ b) = max {a , b} ; (
a ∧ b) = min {a , b} , a , b ∈{0, 1}.
Let be any two boolean matrices of the same type. Find A ∨ B and A∧B.
Let B be the set of all boolean matrices of the same type. We only
state the properties of meet and join.
A, B ∈ B, A ∨ B = [ aij ] ∨ [bij
] = [ aij ∨ bij
]
∈ B.
(Because, ( aij ∨ bij
) is either 0 or 1
∀i ,
j . ∨ is
a binary operation on B.
A∨(B∨C) = (A ∨
B) ∨ C, ∀A,B,C ∈ B. ∨ is associative.
∀A ∈ B, ∃ the null matrix 0 ∈ B ⋺ A ∨ 0
= 0 ∨ A = A . The identity element for ∨ is the null matrix.
For any
matrix A ∈ B,
it is impossible to find a matrix
B ∈ B ⋺A ∨ B = B ∨ A = 0 . So the inverse
does not exist.
Similarly,
it can be verified that the operation meet ∧ satisfies (i) closure property (ii)
commutative property (iii) associative property (iv) the matrix exists
as the identity in B and (v) the existence of inverse is
not assured.
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