Week 8 – SS1 First Term Further Mathematics Notes

 
 WEEK EIGHT:
TOPIC: 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.
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
   1. 2x + y = 3, 4x² – y² + 2x + 3y= 16
   2. 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)