BINARY OPERATIONS: IDENTITY AND INVERSE ELEMENTS

Back to: Further Mathematics SS1

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.

Example: The operation * on the set R of real numbers is defined by a*b = 2a-1┼ b

2

for all a, b € R. Determine the identity element.

Solution:

a*e= e*a = a

a*b= 2a-1┼ b

2

a*e = 2a-1┼ e = a

2

2a-1+ 2e = 2a

2e = 2a-2a +1

e   = ½.

 

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

(a)  2x + y = 3, 4x² – y²  + 2x + 3y= 16

(b)  2^{2x – 3y} = 4, 3^{3x + 5y} – 18 = 0

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

Weekend Assignment

1.     Find the identity element e under this operation if the binary operation* is defined by c * d = 2cd+ 4c+ 3d for any real number.

A. -3                      B. -2C+3       C. X-3

2C+3                                            2C

2.     An operation is defined by x*y = Log^y_x   , evaluate 10* 0.0001

A. 4                       B. -4                C. 3

3.    The binary operation * is defined by x*y= x^y– 2x -15, solve for x if x*2= 0

A.x= -3 or -5         B. x= -3 or 5    C. x = 3 or 5

4.     A binary operation * is defined on the set R of real numbers by

m*n = m + n² for all m, n € R. If k*3 = 7*4, find the value of k

A. 8                      B.28/3             C.14

5     .Find the inverse function a^-1 in the binary operation ∆ such that for all a,b € R

a ∆ b = ab/ 5

A. 25/a                    B.-25/a            C. a/5      

 

      Theory

1. A binary operation * is defined on the set R of real numbers by

x*y = x² + y²+ xy for all x, y € R. Calculate (a)  ( 2*3)* 4

(b) Solve the equation 6*x = 27

2.   Draw a multiplication table for modulo 4.

(b) Using your table or otherwise evaluate (2X3) X (3X2)

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