**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**

- An operation on the set of integers defined by a*b = a
^{2}+ b^{2}– 2a,find 2*3*4 - Solve the pair of equations simultaneously

- 2x + y = 3, 4x
^{2}– y^{2}+ 2x + 3y= 16 - 2
^{2x – 3y }= 4, 3^{3x + 5y }– 18 = 0

** **

**Weekend Assignment**

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

__-3__B. -2C+3 C.__X-3__

2C+3 2C

- An operation is defined by x*y = Log
^{y}_{x }, evaluate 10* 0.0001 - 4 B. -4 C. 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

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

m*n = m + n^{2 }for all m, n € R. If k*3 = 7*4, find the value of k

- 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

- 25/a B.-25/a C. a/5
^{ }

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^{ }** Theory**

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

x*y = x^{2} + y^{2}+ xy for all x, y € R. Calculate (a) ( 2*3)* 4

(b) Solve the equation 6*x = 27

- Draw a multiplication table for modulo 4.

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

See also

OPERATION OF SET AND VENN DIAGRAMS

INDICIAL AND EXPONENTIAL EQUATIONS