Boolean Matrix Zero-one Matrices

1. Boolean MatrixZero-one Matrices • All entries are 0 or 1. • Operations are  and . • Boolean product is defined using:  for multiplication, and  for addition.

2. Boolean Operations

3. Boolean Product A B = Since this is “or’d”, you can stop when you find a ‘1’ A B =

4. Boolean Product Properties • In general, A B  B A • Example B A A B =

5. 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, ∩ , -)

6. 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

7. 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?)

8. 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

9. 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

10. 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) =(AUB) ∩ (AUC) A ∩ (B UC) =(A ∩ B) U (A ∩ C)

11. 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?)

12. 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.

13. 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 □.

14. 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?)

15. 1.6. Mathematical Structures • 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.