Question

Let * be the operation on the set R of real numbers defined by a * b = a + b + 2ab
(a) Find 2 * 3,3 * (-5), 7 * (1/2)
(6) Is (R, *) a semi-group? Is it commutative?
(©) Find the identity element
(d) Which elements have inverses and what are they?
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Answer 1

Answer:

a) =6,-15,3.5

c) 2*3=2+3+2(2)(3)

6=5+12

6=17

=17/6

=2.83

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Answer 2

Answer:

a) 6 , -15 , 3.5

b) (R, *) is a semi group.  It is not commutative

c) The identity element is 0

d) a⁻¹ = -a /( 1+2a)

Explanation:

a) 2*3 = 6

  3* (-5) = -15

  7* (1/2) = 3.5

b) (A , *) is an algebraic system , where * is a binary operation on R , the    the system (A , *) is said to be semi group. To be semi group it should satisfies the following conditions,

1. The operation * is a closed operation on set R

2. The operation * is an associative operation

It is not commutative.

Commutative means that if the order changes and it does not make any differences.

For example, a+b = b+a  ; a*b = b*a

c) The identity element is 0 .

a*e = a+e+2ae

a = a+e+2ae

e(1+2a) = 0

e= 0/(1+2a)

e = 0

d) The inverse element of a is  

a*a⁻¹ = e = 0

a+ a⁻¹ + 2a.a⁻¹ = 0

a⁻¹( 1+2a) = -a

a⁻¹ = -a/ (1+2a)

∦SPJ3