A Boolean Matrix is a real matrix whose entries are either 0 or 1.

**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 = [a_{ij}]
and B = [b_{ij}] be any two boolean matrices of the **same type**.
Then their **join**
∨ and **meet **∧** **are defined as follows:

A ∨ B = [a_{ij}] ∨ [b_{ij}] = [a_{ij} ∨ b_{ij}] = [c_{ij}]

A ∧ B = [a_{ij}] ∧ [b_{ij}] = [a_{ij}
∧ b_{ij}] = [c_{ij}]

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 *=* *[* a _{ij} *]

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.

** **

Tags : Discrete Mathematics | Mathematics , 12th Maths : UNIT 12 : Discrete Mathematics

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12th Maths : UNIT 12 : Discrete Mathematics : Some binary operations on Boolean Matrices | Discrete Mathematics | Mathematics

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