Binary Operation

Binary Operation

  • Definition: A binary operation on a set ( S ) is a function ( * : S d7 S 7; S ) such that:
    • For every pair ( (a, b) ) of elements in ( S ), ( a * b ) is defined and there exists a unique element in ( S ).
Conditions to Show Binary Operation
  • To confirm whether ( * ) is a binary operation, two conditions must be satisfied:
    • Closure Property: For any two elements ( a ) and ( b ) in the set, ( a * b ) must also be an element of ( S ).
    • Uniqueness: If ( a * b = c ) and ( a * b = d ), then it must be true that ( c = d ).
Examples of Binary Operations

  1. Addition: Let ( S = \mathbb{R} ) (the set of real numbers) and the operation ( * ) be defined by ( a * b = a + b ).

    • Closure: For any ( a, b \in \mathbb{R} ), ( a+b \in \mathbb{R} ).
    • Unique: Addition is unique for real numbers.

  2. Multiplication: Similar to addition, with ( a * b = a \cdot b ).

    • Similarly satisfies closure and uniqueness.

  3. Maximum Operation: Let ( S ) be the set of real numbers and the operation ( * ) be ( a * b = \max(a, b) ).

    • Satisfies both closure and uniqueness properties.

  4. Division: Let a * b = a / b.

    • Not a binary operation:
      • Undefined for ( b = 0 ).
      • Does not satisfy closure (e.g., ( 1 / 0 ) is undefined).
Properties of Binary Operations

  1. Commutative Property:

    • A binary operation is commutative if ( a * b = b * a ) for all ( a, b \in S ).
    • Example: ( a + b ) is commutative, but ( a \cdot b ) is not.

  2. Associative Property:

    • A binary operation is associative if ( (a * b) * c = a * (b * c) ) for all ( a, b, c \in S ).
    • Example:
      • For Addition: ( (a + b) + c = a + (b + c) ) is true.

  3. Identity Element:

    • An element ( e ) in ( S ) is an identity for ( * ) if ( e * a = a * e = a ) for all ( a \in S ).
    • Examples:
      • For addition: Identity is 0. ( a + 0 = a ).
      • For multiplication: Identity is 1. ( a imes 1 = a ).

  4. Invertible Element:

    • If ( e ) is the identity and for every ( a ) in ( S ), there exists an ( b ) such that ( a * b = e ), then ( a ) is invertible (with ( b ) being the inverse of ( a )).
Constructing a Binary Operation Table
  1. Step 1: Write all elements of the set in the first row and first column.
  2. Step 2: Perform the binary operation for each pair and fill in the table.
Example of Binary Operation for Small Sets

  1. Multiplication Table for Set ( S = {-1, 0, 1} )

    *-101
    -110-1
    0000
    1-101

  2. Modulo Operation:

    • Let ( S = {0, 1, 2, 3, 4, 5, 6} ) and define ( a + b \mod 7 ).
    • Example:
    • Construct the addition table modulo 7 where the sum cycles back every 7.
Conclusion
  • Binary operations are fundamental in mathematics and can be characterized by properties such as closure, commutativity, associativity, identity, and invertibility, as shown in various operations including addition, multiplication, and custom-defined operations.