Binary Operations: Identity And Inverse Elements

Identity element and Inverse element

CONTENT:

Identity Element:

     Given a non- empty set S which is closed under a binary operation * and if there exists an element e € S such that a*e = e*a = a for all a € S, then e is called the IDENTITY or NEUTRAL element. The element is unique.

Evaluation

Find the identity element of the binary operation a*b = a +b+ab

Inverse Element;

     If x € S and an element x^-1 € S such that x*x^-1 = x^-1*x= e where e is the identity element and x^-1 is the inverse element.

Example: An operation * is defined on the set of real numbers by x*y = x + y -2xy. If the identity element is 0, find the inverse of the element.

Solution;

      X *y = x+ y- 2xy

      x*x^-1 = x-1*x= e, e = 0

      x + x^-1– 2xx^-1 = 0

      x^-1 -2xx^-1= -x

      x^-1(1-2x) = -x

      x^-1 = -x/ (1-2x)

The inverse element x^-1 = -x/ (1-2x)

Evaluation:

 The operation ∆ on the set Q of rational numbers is defined by: x∆ y = 9xy for x,y € Q

 Find under the operation ∆ (I) the identity element (II) the inverse of the element a € Q

General Evaluation

1. An operation on the set of integers defined by a*b = a² + b² – 2a,find  2*3*4
2. Solve the pair of equations simultaneously
3. 2x + y = 3, 4x² – y²  + 2x + 3y= 16
4. 2^{2x – 3y} = 4, 3^{3x + 5y} – 18 = 0

Reading Assignment: Read Binary Operation, Further Mathematics Project II, page 16 – 22