B tech. Discrete mathematics (I. T & Comp. Science Engg.) Syllabus



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SEMIGROUP

A non-empty set S together with a binary operation  is called as a semigroup if –



  1. binary operation  is closed

  2. binary operation  is associative we denote the semigroup by (S, )


Commutative Semigroup :- A semigroup (S, ) is said to be commutative if  is commutative i.e. a b b aa S

Examples : 1) (z, +) is a commutative semigroup

  1. The set P(S), where S is a set, together with operation of union is a commutative semigroup.

  2. (Z, –) is not a semigroup

The operation subtraction is not associative

IDENTITY ELEMENT :

An element e of a semigroup (S, ) is called an identity element if e a a e a a S


Monoid A non-empty set M together with a binary operation *defined on it, is called as a monoid if –


  1. binary operation  is closed

  2. binary operation  is associative and

  3. (M, ) has an identity.

i.e. A semi group that has an identity is a monoid.
A a non-empty set G together with a binary operation  defined on it is called a group if


  1. binary operation  is close,

  2. binary operation  is associative,

  3. (G,  ) has an identity,

  4. every element in G has inverse in G, We denote the group by (G,  )




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