Algebra

Definition: A non-empty set G having binary operation “*” (say) is called group in math if it satisfies the following axioms: i.e, . There is identity element e in G such that . For each , there is an such that you can also check semi group and monoid

Completing the Square examples: Sometimes, the quadratic polynomials are not easily factorable. For Example, consider It is difficult to make factors of In such a case the factorization and hence the solution of a quadratic equation can be found by the method of completing the square and extracting square roots. Completing the Square Examples Say

Order of a Group: The order of group  is the number of elements present in that group , also say it’s cardinality. It is denoted by or . Examples: Dihedral group has order Symmetric group has order ={} modulo has order . Order of an element: Order of element is the smallest positive integer such that , where

Definition: A non empty set G having binary operation “*” (say) is called Monoid if it satisfies the following axioms: Closure Property w.r.t “” i.e, Associative Law w.r.t “” Identity element exist. There is identity element e in G such that If the above three properties hold then the set is called Monoid. We can

Definition: A non empty set G having binary operation “*” (say) is called semi group in math if it satisfies the following two axioms: Closure Property w.r.t “” i.e, Associative Law w.r.t “” Examples of Semi Group: The set of W- {0,1} with respect to “Addition” and “Multiplication”. The set of Natural Number with respect to “Addition”. The