SUMMARY ( ) A binary operation * on a non-empty set S is a rule, which associates to each ordered pair a b of elements a b , in S an unique element a b * in S . ( ) Commutative property: Any binary operation * defined on a nonempty set S is said to satisfy the commutative property, if a b b a a b S ∗= ∗ ∀ ( ) Associative property: Any binary operation * defined on a nonempty set S is said to satisfy the associative property, if a b c a b a b c S ∗ ∗ ∗ ∗ ∀ , , ( ) Existence of identity property: An element e S Î is said to be the Identity Element of S under the binary operation * if for all a S Î we have that a e ∗= and e a ∗ ( ) Existence of inverse property: If an identity element e exists and if for every a S Î , there exists b in S such that a b ∗= and b a ∗ = then b S Î said to be the Inverse Element of a . In such instance, we write b − . ( ) Uniqueness of Identity: In an algebraic structure the identity element (if exists) must be unique.
( ) Uniqueness of Inverse: In an algebraic structure the inverse of an element (if exists) must be unique. ( ) A Boolean Matrix is a real matrix whose entries are either or . ( ) Modular arithmetic: Let n be a positive integer greater than called the modulus . We say that two integers a and b are congruent modulo n if b − a is divisible by n .
In other words ≡ (mod n) means a n k ⋅ for some integer k and b is the l east non-negative integer reminder when a is divided by n . ( ( ) Mathematical logic is a study of reasoning through mathematical symbols. ( ) Let p be a simple statement. Then the negation of p is a statement whose truth value