# BINARY OPERATIONS: IDENTITY AND INVERSE ELEMENTS

## 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
• 2x + y = 3, 4x2 – y2 + 2x + 3y= 16
• 22x – 3y = 4, 33x + 5y – 18 = 0

## 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.
1. -3                   B. -2C+3       C. X-3

2C+3                                         2C

1. An operation is defined by x*y = Logyx   , evaluate 10* 0.0001
2. 4 B. -4                C. 3
3. The binary operation * is defined by x*y= xy– 2x -15, solve for x if x*2= 0

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

1. 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

1. 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

1. 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

1. Draw a multiplication table for modulo 4.

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

BINARY OPERATIONS

OPERATION OF SET AND VENN DIAGRAMS

BASIC CONCEPT OF SET

LOGARITHM

INDICIAL AND EXPONENTIAL EQUATIONS

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