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## Algebraic Structures

The algebraic structures like semigroup, monoid, group, ring and field have wide applications in many disciplines and in particular to binary coding.

##### 5.1 BINARY OPERATIONS

Definition 5.1

A collection of objects with operations defined on them and the accompanying properties form a Mathematical Structure or System.

For example, the collection of sets with the operations of union, intersection, and complement and their accompanying properties is a discrete mathematical structure. We denote this structure by [sets, ∪, ∩, −].

Similarly, the collection of 3 × 3 matrices with the operations of addition, multiplication and transpose and their accompanying properties is a mathematical structure. We denote this structure by [3 ...

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